Fixes & Explanations

Quicksilver and Banshee in X-Men Origins 

Yes. They are supposed to be Quicksilver and Banshee and they are meant to become X-Men along with Cyclops and Silver Fox. They are clones of the original mutants. We don't truly know what happened to them in Timeline Sigma anyway. 

Knowledge about mutants

Another contradiction which is seldom brought up actually, but which is quite significant. In X1, we're led to believe that the mutant phenomenon has only recently become a significant topic of public discourse.
In First Class, we learn that the existence of mutants was revealed to the world way back in 1962.

As we already know, First Class never happened in the original timeline, Timeline Prime Alpha. So X1 doesn't "follow" First Class and its events.

We need to remember that in First Class, its basically the US and Soviet governments which learn off the existence of mutants, not the general public. Its possible that the government kept the existence of mutants a secret, even as people like Stryker were secretly recruiting them for black ops missions and later experimenting upon them. Origins suggests that the public at large is unaware of the existence of mutants, even though the government is well aware of them. So its possible that for decades, mutants were an urban myth, like alien abductions and stuff, but by the time of the trilogy, the cat was out of the bag and it became a full-blown "mutant phenomenon".

After Bolivar Trask is killed, the one man who was pretty much awake for all of it was Stryker. I think that developed an interest in mutants and then when his son came along he sent him to Xavier for help but Xavier could not do anything. So instead of having a general interest he starts hating them and starts experimenting them.

DOFP also states that a tenth of a tenth of a tenth of a percent of people are mutants at that time - 1973. 0.001% of approximately 210,000,000 or maybe a little over 2,100 people in the United States.
Even if that grew exponentially throughout the next 20 years. I would say it wouldn't be until around 2004 that we would see major conflicts in these areas.  

 

Why is Xavier alive in DOFP (2023)? He has been killed by Phoenix in The Last Stand!

"P." Xavier is the supposed brain-dead twin of Charles. Even though there is no official record of him, it may imply that he was born brain-dead due to Charles' mutation in his mother's womb as the latter's telepathy may have been involved in some way. According to Charles: "The man you see here was born with no higher-level brain functions. His organs and nervous system work, but he has no consciousness to speak of".  

Because of his condition, he was always moved from one hospital to the next until the Muir Island facility was constructed and Charles sent his twin under Moira McTaggert's care.

In the post-credits scene of The Last Stand, it is implied that Xavier has managed to transfer his consciousness into the body of his brother. He cannot walk anyway because "P." Xavier's legs have been permanently damaged/atrophied by the 68 years-long coma. Finally, the man looks identical to Xavier (but sports a long beard).

  

Why Scott Summers' powers are slightly different in Origins? 

They are not different. Friction generates heat. His optic beam is pure concussive force coming from an alternate dimension (his eyes work as an interdimensional portal). As such, that "force" can generate heat in special occasions, as byproduct of friction. Anyway, we can just see few tiny flames around the edges of the wreck he caused in the school, while Deadpool's optic beam (genetically modeled after Cyclops' one) managed to overheat Wolverine's claws. 

How much old is Beast in The Last Stand?

We must assume his enhanced mutation slowed down his aging process.
There is no proof to conclude that he doesn't have the ability to revert to a human appearance in The Last Stand, he may just elect not to. Or it could be a case of needing to continually take injections to have that ability and since in his older age he no longer feels shame about the way he looks, decides to stop taking the serum. And if that were true, it would explain his human appearance in X2.

Was Cyclops in First Class? So was him a teenager at the time (1962)?

No, not at all. Those were "easter eggs" for the fans.

Magneto, Mystique, Professor X, Cyclops, Beast and all others look very healthy and younger than their anagraphic ages in the original trilogy and DOFP. How's that ever possible?

Generally speaking, cinematic mutants' aging process is slightly slowed down by default. The Gene X may improve the biological process; it's a sort of optimization. Mutants age slower and better than humans. It's pretty established that Beast and Mystique are aided by their specific mutations, even.

How can adamantium cut/penetrate adamantium? 

This is not comic book adamantium. They created new rules for its cinematic counterpart. First of all, it's an alien metal (like vibranium in the comic books). Summing it up: if given a mild, specific overheat signature, hot adamantium can break cold adamantium (Wolverine's claws pitted against Deadpool in Origins; Silver Samurai's katanas in The Wolverine). If submitted to extreme and prohibitive hot temperatures, then adamantium becomes "fluid" and "malleable", as established in X2. In regards to the "adamantium bullets" in Origins, we can explain that by saying they were provided with an overheat signature, exactly like Silver Samurai's katanas.

Why did Logan's adamantium-laced claws look so shining in some scenes of Origins? 

You can assume the adamantium covering his bone claws was still fresh and not "worn out" as later in the saga. Plus, it's an alien metal, so you can suppose this substance just looked more "outerworldly" (even shining) in its newborn stadium.

And now, the big question. At the end of The Wolverine movie we see Logan's adamantium-laced claws getting cut off and his natural bone claws coming back. In DOFP is there an explanation as to why he gets his adamantium claws back?

It's very subtle. Magneto is an X-Man. There's a strong hint he could "perform" a molecular binding between his bone claws and metals in DOFP. It was young Magneto to suggest that, but it's still valid for old Magneto, of course.