blitz:
1) You could use the 3.3v to power something else... an LED, another PCB, and so on.
2) If it's a single common PCB, you only need to connect to one of those "ground" points for all the buttons, IF the board truly has a single common (more on that at the end)
3) L3 and R3, clicking the analogs in. Look at where they are in relation to the analog sticks on the other side of the board, it'll make sense.
4) Because triggers cannot simply be shorted to common to work. They use variable resistors, and will require knowledge of the circuit and/or additional resistors to get it working with a standard arcade pushbutton.
5) This ties into 4. If you're going to leave the triggers intact, the trigger needs to be seen as "disengaged" to the 360. If you remove all the trigger plastic (VERY difficult on a Microsoft pad), you are still left with a soldered potentiometer. From here, you have options, depending on what you need/how lazy you are. You could set it to wherever it was at when the trigger mechanism wasn't pressed, you could set it there and hot-glue it in place, or you could remove it and use resistors to imitate that "neutral" setting on the potentiometer.
Okay, with all that said, if you don't have a multimeter, get one, and test these commons yourself. I am grateful to slagcoin for all his diagrams, but I have found some of them to be wrong, and it's best to do your own testing anyway. If the board has a single common, all the common points should all show the same level of continuity when you test between them.
Also, if you're thinking of using the triggers, be VERY careful removing them. It is VERY easy to break the transistors off of a 360 PCB, which will ruin the board's functionality. Soldering them back on is pretty futile, because the copper will usually be pulled off the board as well. Yes, I've done it, and I'm stupid. Removing them entirely will probably mean carefully cutting them away.