I have yet to see a quote specifically stating that the indoctrination theory is incorrect. If you could provide a link to that, it'd be appreciated.
Personally, I think the indoctrination theory is a little more believable than taking the ending at face value. Either is possible, but both have severe issues attached to them.
1. If what you see is really what happens, then there's more plotholes than can honestly be believed. The things occurring on-screen simply do not make sense, no matter which way you swing it.
2. If the indoctrination theory is correct, then the plotholes become irrelavent as it hadn't actually happened. However, this means that Bioware/EA deliberately left out a proper, conclusive ending to either create another sequel, or DLC to expand the ending, either through the cutscenes, or some small amount of gameplay. We can all try to look at what Bioware said about the upcoming extended ending DLC, but anything about their plans could change.
I could get into a pile of little details about how much really is broken about the ending, but all of that information is found in the AngryJoeShow video, and ACAVYOS' indoctrination theory video.
Now, onto direct responses to the spoiler-talk:
@42:25: Questioning the "new" ship in ME3
When ME2 ended, provided Shepard survived the suicide mission to the Collector Base, the Alliance grounded him and began proceedings for a trail against Shepards' works with Cerberus (or the destruction of the Batarian-run system, more importantly, if the Arrival DLC was completed in ME2). While grounded on Earth, the Alliance military/government took the Normandy SR2 apart, examined it for information about Cerberus and their technology, then rebuilt it with minor upgrades and an Alliance-based branding. Any major upgrades that were made to the ship in ME2 were kept, but mildly improved in case it needed to be used again. If you explore the Normandy throughout ME3, talk to specific crew members, and use the Spectre Terminal in the offices on the Citadel to read through/approve specific events, you could upgrade certain parts of the Normandy further, though these were mostly for a lengthier interaction with the crew members.
As for Shepard keeping the ship, Cerberus acknowledged from the beginning that Shepard was in charge, and that it was his ship, but more incentive for him to work with Cerberus, as opposed to directly working against him. The Illusive Man knew that he needed to give Shepard reasons to help Cerberus, so he catered to Shepards' sense of connection to his team and ship in order to gain Shepards' skills.
@44:10: Gender-based decision on Ashley/Kaiden
Someone else has already mentioned this, but depending on which choice you make on Virmire in ME1, either Ashley or Kaiden will survive. Say, for example, that Kaiden is helping Captain Kirrahe to repel the geth forces, while Ashley ends up needing to set up the bomb. If you choose to return to the bomb, Ashley will be saved, but if you choose to continue pressing forward towards Kirrahe and Kaidens' team, then Kaiden will live. Gender has nothing to do with the outcome.
@46:35: Shepards' fate at the final decision
It's certainly not clear as to why Shepard needs to die while choosing to control, destroy, or synthesize the Reapers. However, it makes a bit of sense if you think about each scenario. Destroy, everything around Shepard is exploding, likely engulfing him (unless you get the breathing scene on Earth, which requires over 5000 Effective Military Strength). Control and Synthesize, the energy essentially disintegrates his body, using his energy to, respectively, control the Reapers or Synthesize all life to be equally organic and synthetic. A crazy explanation/observation one way or another, but it at least makes some kind of sense to me.
@47:40: The identity of the Catalyst
This sort of ties in with Shipwrecks' mentions of the indoctrination theory. The Catalyst, as established before the final assault to reach the Citadel from Earth, was the Citadel itself. This was undisputed and made some amount of sense, though it was somewhat unexplained. However, upon reaching the final decision once the Illusive Man and Anderson had died, this odd ghostly apparition of the child Shepard saw die on Earth suddenly reveals itself to be the Catalyst.
I see two ways of thinking of this; either this odd character really is the Catalyst, and the Catalyst was misinterpreted to be the Citadel, or this "Catalyst" is actually a relatable figure that Shepard can recognize and sympathize with, created by the Reapers through the indoctrination process in order to force Shepard to choose what they desire. If this final sequence did take place in Shepards' mind and was just an internal battle between Shepard and the Reapers to fight off indoctrination, it would make sense that the Reapers use that child, who Shepard regrets not being able to save, in order to sway his decisions in their favor (either the Control or Synthesize option, relating to the Illusive Mans' or Saren's desires, both of which would ultimately lead to the Reapers gaining what they want, either through ruling all synthesized beings, or breaking free of Shepards' control).
@49:18: Wombat's number of endings
As far as I know, there are 6 variations of endings. 1 for Synthesize, 2 for Control, and 3 for Destruction. Synthesize is always the same: everything becomes part-organic-part-synthetic, mass relays destroyed, Shepard died. Control has two outcomes: the Reapers leave OR the Reapers leave and some stuff is mildly destroyed, mass relays destroyed, Shepard dies. Destroy has the most variation, which still isn't saying much: 1, the "Vaporize" variation, where everything the red explosions touch is simply destroyed. Reapers/all other beings alike are subject to this fate, mass relays destroyed, Shepard dies. 2, the Reapers are destroyed, some damage done to Earth, mass relays destroyed, Shepard dies. 3 (this is obtained by having over 5000 Effective Military Strength), the Reapers are destroyed, "minimal" damage to Earth is sustained, mass relays destroyed, and Shepard lives.
This final of the 6 endings if the ONLY ending variation to show the breath scene, indicating something more to that particular branch of story. What this really means isn't quite clear, but it's been used to support the indoctrination theory: that the entire final sequence after being knocked out by Harbingers' attacks was in Shepards' mind, hence the "waking up" after defeating the indoctrination process. That would mean that the entire battle to save Earth is still on-going, and Shepard really does still need to complete the mission. If the indoctrination theory is incorrect, well, this leaves much to be explained, as it was very clear in ME2 that Shepard falling from space to a planet KILLED HIM. How he would have survived the fall from the Citadel and Crucible back to Earth without being burnt to a crisp upon re-entry simply wouldn't make sense.
As a personal aside, considering the extra breath scene can only be scene by choosing Destroy with over 5000 Effective Military Strength, it makes me believe that, ultimately, Bioware/Casey Hudson is attempting to create a canonical scenario for the universe, which makes many players' stories completely moot for the inevitable continuation (in some form) of the Mass Effect franchise. This honestly makes me more mad than anything else in regards to the ending. I'm fine with having a canonical scenario to a grand story, but when the franchise is built off of the specific decisions each player makes, it's hard to believe that any sort of canon outside of character, race, and history lore can remain consistent for long.
@54:46: "What the hell has been solved?"
if the indoctrination theory is being followed, then Shepard "waking up" on Earth after the final sequences indicates that Shepard has overcome the Reapers' indoctrination process. He would have successfully chosen, for himself, in his mind, not to succumb to the Reapers' will, by deciding to destroy them, rather than try to control or change them. Ultimately, in reality, it means that nothing has been solved, and that Shepard still needs to finish the battle against the Reapers.
Something to note, that ACAVYOS points out in his indoctrination theory video, is that when you choose either Control or Synthesize, there is a distinct blue glow to Shepards' eyes, something both the Illusive Man and Saren sported while under the influence of the Reapers. The only ending where this is not present is Destroy, and all 3 ending types make a point of very clearly giving the player time to notice Shepards' eyes. All variations focus on his eyes, or at least face, for a time, giving you a chance to notice the differences.
@56:25 - 56:40: The mass relays
One of two things can be thought of with the destruction of the mass relays. One, everyone is now stranded where they are, and are forced to travel at much slower speeds, making space travel over long distances extremely inefficient. Or, two, the information from the ME2 Arrival DLC is noted, taking into account that destroying a mass relay DESTROYS THE ENTIRE SYSTEM IT RESIDES IN. This essentially makes every single ending lore-breaking, unless someone tries to say that the use of the Crucible somehow made it a "controlled" demolition of the relays, as opposed to Arrivals' smashing an asteroid into the relay. One way or another, there's a massive explosion of each relay, as seen during the destruction of the first one in the final cutscenes, and honestly, it doesn't look like a positive, controlled explosion. The way the relays were handled overall, regardless of opinion, was poor, and there's little way to explain what happened. The only way to avoid believing the relays' destruction isn't screwing everyone over is if the indoctrination theory is believed, and not everyone will believe that theory. This is going to create a HUGE problem for Bioware in trying to appease people with the upcoming "extension" DLC. The endings will be vastly different for everyone, yes, but as I've previously mentioned, the canon of the universe will constantly be in question.
@1:01:35: EDI's 'working parts'
EDI's body was Dr. Eva Core, who was designed and created in a manner similar to Project Lazarus, the project that was designed to revive Shepard as he was before his early ME2 death. Eva Core was meant to essentially replace Shepard as the Illusive Man's greatest asset, and is essentially the Mass Effect equivalent of a 'fembot', a fully functional woman-like robot that just so happens to not actually be a living being.
I should also note, for anyone that may have noticed, that this Eva Core is NOT the same Eva Core as seen in other Mass Effect mediums and stories. The Eva Core from the readable material is the Illusive Man's (real name, Jack Harper) friend and teammate from during the Battle of Shanxi in the First Contact War. The 'fembot' known as Eva Core in ME3 is named after Jack's/Illusive Man's friend, who died on Palaven while attempting to free Ben Hislop (another teammate who was controlled by a Reaper artifact) and achieve revenge on Desolas Arterius, Sarens' brother, whom they all fought against on Shanxi.
@1:01:45 - 1:02:05: Mordin
While most people won't have found this (I certainly wouldn't have sacrificed so much to do it), it is possible to save Mordin from dying. In most cases, Mordin will attempt to cure the genophage by reaching the top of the Shroud, and ultimately sacrifice himself in the process, whether or not the cure is successfully dispersed. However, in very certain conditions, he can survive, but at a great loss/feeling like a total dick. Wrex must be dead (making Wreav the new leader of Clan Urdnot and rallying all of the krogan), Maelon's data on the genophage from Mordin's loyalty mission in ME2 must be destroyed (or the mission avoided altogether), and you must not tell the clan leader, Mordin, and Eve about the Dalatrass' blackmail and prior sabotage of the Shroud. Doing all of this leading up to the moment when Mordin tells Shepard he needs to reach the top of the Shroud will allow for a rare possibility. You must attempt to stop Mordin from going by choosing primarily Renegade options, ultimately leading to Mordin realizing that the Shroud is sabotaged. If Shepard can successfully convince Mordin not to cure the genophage, you will save Mordin from sacrificing himself and receive a larger war asset count from the Salarians, but at the cost of the krogan being doomed to suffer the genophage (and already having lost Wrex as a team member, war asset, and proper (mostly) level-headed leader for the krogan).
On a personal note... Mordin's death was one of the most depressing things I've ever seen. I didn't get the seashell line or the song (I now love 'Salarian Scientist'. XD), but it was supremely sad regardless.
@1:07:50: Any other Spectres?
There are very few other Spectres that have been seen or mentioned, but there are, certainly, others.
In ME1, Shepard, Saren, and Nihlus are the only known Spectres, and both Nihlus and Saren are dead by the end of the game.
In ME2, Tela Vasir, the Shadow Broker assassion in the Lair of the Shadow Broker DLC, is a Spectre using her resources to assist the Shadow Broker in taking out Liara. Tela also, ultimately, dies, before assaulting the Shadow Broker's base. Another is mentioned in a Cerberus news report, Lonar Maerun, but is never shown; the news report was dated December 23rd, 2010, taking place well before humans even discovered the Prothean ruins on Mars.
In ME3, either Ashley or Kaiden will become a Spectre (part of Udina's plot against the Citadel with Cerberus), and you also run into Jondum Bau, the salarian Spectre working to track down a reportedly indoctrinated hanar diplomat (this is the mission where you meet Kasumi again).
I've also heard that, if Shepard refuses the offer of Spectre reinstatement, that eventually Captain Bailey will mention that only two individuals in the past 1000 years ever refused the offer to be a Spectre, both of which were asari matriarchs. Makes me wonder if Samara and Benezia were offered the position at some point, or if one of the other mentioned Matriarchs (Aethyta, Dilinaga, etc.) were offered the part.
@1:15:25 - 1:15:38: Galactic Readiness and Hammer
Galactic readiness does affect very little, but having 5000+ Effective Military Strength is the only way to have a chance at seeing the breath scene during the ending. This essentially makes it crucial to play either the multiplayer portion of the game or the iPhone game Mass Effect: Infiltrator, or the information center Mass Effect Datapad.
Now, the deal with Hammer was that they were part of the three-tiered team to deal with the fight to take back Earth and defeat the Reapers. It was mentioned prior to the attack, mostly referring to the plans for the Crucible. The Sword team were the attacking squads, acting as a distraction for the Reapers while Hammer, the ground forces, would land on the planet and make their way towards the beam that lead to the Citadel. Shield was the defense, the team protecting the Crucible on its' way to the the Citadel.
This was a great show, and I was happy to hear more ME talk after listening to the Rooster Teeth spoiler-cast prior. I really liked the new game ideas you all came up with to extend the universe, espectially the Spectre-recruiting idea Wombat pitched. What would you all think about a prequel game where you played as the Protheans prior to the end of their cycle, or a game taking place during the First Contact War? Would these stories be better suited for other media? (The First Contact War has already been somewhat touched on in the comics, revealing a sort of origin story on the Illusive Man, but I don't know how much of the war was expanded on.)