*Dats the power of Geass for you. The power of the Human Subconscious can do that for you. There is even a power that gives the wielder the ability to shapeshift.
*Got sent away because their other family members might target them due to their line of succession. Charles witnessed it himself and did not wish it to happen anymore. Lelouch and Nunnally were apparently the favored children of Charles, largely due to his favorite consort being Marianne. As such, that would make the two ideal targets for their siblings if Lelouch or Nunnally were named successor.
*Suzaku is siding with Lelouch because Lelouch promised he would die and atone for his sins, especially for accidentally getting Euphie killed and ruining her reputation. It also was intended to bring about a revolution that Suzaku originally wanted.
*Nunally sorta believed in Prince Schneizel's plan and wanted to make the Damocles a symbol of hatred and allow them to move forward in life after it is gone. It goes against Lelouch's plan since he wants Him to be the symbol, not the Damocles. Nunally said she hated Lelouch because he is not the brother she used to know.. Even though her eyesight returned, she still believes she is blind. Yes. Nunally was aware she was of Royal Blood. But she didn't care about it since she only wanted to live with Lelouch.
*Lelouch was forced to be the bad guy due to the current state of the world. A large amount of distrust would exist and there would still be conflict. Lelouch believed that if the world focused all their hatred on him, it would unify the world and bring about peace once it is gone since at that point, people would be tired of conflict and would want to focus on rebuilding and bringing about peace. Lelouch did join the United Federation. Then he screwed them over.
*Britannia really wanted to conquer Japan because the country had the largest deposits of Sakuradite in the World. Sakuradite is the fuel for Knightmare Frames among other things (their warships, weapons, etc). Britannia didn't conquer the rest of the world at that point. They barely managed to conquer the European Federation at S2 I think and the Chinese Federation was still hanging on. Britannia exerted control over their conquered territories by appointing a Grand Duke, as seen in Akito the Exiled. The Grand Duke basically served as a proxy Ruler who only answered to the Emperor. The School is in Area 11 through the efforts of the Ashford Foundation, a philanthropic organization involved predominantly in the supply of educational services.
*Lelouch loved Kallen and was aware of her extreme loyalty to him. She would go to the depths of hell if Lelouch commanded her to. He didn't want to see her die/go to her death. And it was for that exact reason, he made her choose between him and her sense of justice. Her sense of justice won.
*C2 only cared for herself. But as time passed on, she eventually warmed up to Lelouch. And once Lelouch told C2 that he cared for her and didn't blame for everything that happened, she regained her "Human" self as well as the will to live and move forward in life even though she is immortal.
*The Black Knights are stupid, especially Ohgi. Ohgi technically

ed everything up since beforehand, things were going the right direction until they forced Lelouch's hand to create the Zero Requem. If the Black Knights did not betray Lelouch, there would've been a more "happy" ending. The fact that the Black Knights took Cornelia's words, an enemy they had been fighting for 1 season and a half as truth immediately made them lose all credibility. Then they decided to shoot first and ask later, ruining any chance of Lelouch fixing things. Basically, The Black Knights turned on Lelouch because they thought he used Geass on them. Of course, part of this is Lelouch's fault since he didn't mention any of this to his Inner Circle as a heads up. They were also less inclined to trust him due to Lelouch abandoning them at the end of S1.
*Sayako is loyal to Lelouch because she served as the sibling's caretaker when they came to the Academy. Sayako's family is employed by the Ashford family and are loyal to them.
*Ashford Academy was owned by the Ashford Family, a Lesser Noble Family loyal and associated with Marianne. When they took Lelouch and Nunnally in, it wouldn't be hard think that the Family used their connections to hide the identities of Lelouch and Nunnally.