Question: Why is Charles okay with Mystique joining Magneto's side? They're supposed to be brother and sister but he doesn't even attempt to talk her out of it.
Answer: He's not okay with it, but he knows that Erik's views appeal strongly to her. To all intents and purposes, she's already made her decision - to try to stop her would likely only lead to an argument, which is not how he'd want them to part. Plus there is the somewhat major side issue that Charles has just been shot, so it's not really the right time for a discussion on conflicting mutant philosophies.
Question: How come Professor X was so against Magneto killing but had no problem with Wolverine killing in other movies?
Answer: In this film Xavier was young and naive, and only met Wolverine once. After, as seen in Days of Futures Past, he's become cynical and hateful towards everything.
Question: Charles amply demonstrated that he could remotely control the mind of a Russian naval officer, causing him to act against orders. However, on the beach, when they are unable to contact the American and Soviet fleets by radio, why didn't Charles simply control the minds of the naval commanders and stop the naval attack?
Answer: He was much closer to the Russian officer in comparison to when he was on the beach, and the Russian officer was just one man. Taking complete control of the hundreds of men on the ships required to fire all those guns would be beyond even Xavier.
Question: Why did Erik despise humans so much, even though it was a mutant who killed his mother?
Answer: Because it was a human who started the Holocaust and got Eric and his family put into a concentration camp in the first place. Seeing how one human reacted to just another race he saw what humans in power would do to another race or species the feared.
Question: When Charles goes to meet the CIA to explain about mutation and mutants, he speaks to Agent Stryker, who is the father of William Stryker. William Stryker gave Logan adamantium which increased his powers. If Charles met William Stryker's father at a young age then how did he meet Logan also at a young age ?
Answer: Logan is around 100 years old at this point. He aged at a normal rate to his peak maturity, then his healing power kept him from aging any further.
Question: If Charles Xavier and Mystique were like brother and Sister, why did he never mention this in the first 3 X-men films?
Answer: Why would he? It is hardly unusual for a person to never mention a past relationship with someone if they were particularly hurt, angered, or disappointed, resulting in them acting as if that person never existed. It can often come as a surprise to others to learn that a friend or relative had some prior relationship they never knew about.
Question: How did Moira, Charles and the others get away from the island? They did not have someone who can teleport them away, the jet is broken and they had no possibility to communicate with the others. You can hardly expect the Russians or the Americans, who wanted to kill them a few minutes ago, to suddenly want to help them.
Answer: They do have the ability to communicate with the other mutants, via the powerful telepath Charles Xavier. He could also use mind control on the navy captains to make them rescue them, or any number of other possibilities.