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: 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.
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.