Answer: The short answer is events in this film negate what happened in X-Men 3 as well as Origins: Wolverine or one could say this film essentially became a reboot. Since no real answer seems to be given, and since the X-Men 3 film makers didn't know Charles would later be shown to become paralyzed prior to visiting Jean, they had no need to explain why he's walking. One can only speculate on the possible ways Charles walks in these 2 previous movies (and this isn't taking into account the timeline shift from X-Men Days of Future Past). We do know from DOFP that Hank/Beast created a serum for Charles that allowed him to walk, albeit without his powers. Hank could have kept working on this serum which would allow Charles to walk and still maintain some of his powers. Then at some point Charles stopped taking the serum, confining him to the wheelchair once again. Either because they ran out of the serum or because Charles came to the realization he needs to accept what happened and not hide it, especially if he's teaching children to accept who they are. Charles also has the power of "astral projection" and the power to appear in the minds of others, so Charles could simply not be there at Jean's house, but is back home, in his wheelchair.
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: The Russian captain commends the naval soldier for "averting a nuclear war" after Charles took over his mind and destroyed the ship with the nuclear weapons. This sounds like a dumb question, but if he was commended for averting nuclear war, why did he throw him in the brig?
Answer: Because he still did it on his own, against orders. Though the captain was happy he did it, he still needs to punish him for disobedience.
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: 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.
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: 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: 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.