Plot hole: In the opening scene, Wade applies to be an Avenger, and does it in the Sacred Timeline, the main MCU timeline, on Earth-616. Not his own, which the movie classifies in the next scene as Earth-10005. Assuming he can just do that (at the end of his second movie he certainly did stranger things than that with Cable's device), one fails to see the purpose of it. Let's assume he aced the interview and got hired; it's not his timeline. He does it to have a good relationship with his girlfriend. Who...is in a different universe. Whatever Wade does in the vastly different universe is not altering anything in his own. It was never going to work.
Suggested correction: He goes back to his universe. Think Thor...or the Guardians or Captain Marvel or whomever isn't Earth based. They "Assemble" then go back to protecting their little corner of the Marvel Universe.
Suggested correction: But it's Deadpool. When does anything he ever does make sense? He wanted to be an Avenger, he went out to be one, no matter what that meant in the literal sense. As long as he can say it, it's enough for him. Even if it is in a different universe/timeline, he can still say he's an Avenger. He is also a rule breaker, so maybe he can figure out how to make it work.
I knew this was going to be the objection to it; it's Deadpool, he can do whatever, etc. There are multiple times during the movie when he mentions things that he is not supposed to know because they are meant to be fourth-wall-breaking jokes. He references real-life actors, he interacted with them in previous movies. So they are absurdities with a purpose, and it's pointless to argue with comedy. But him being rejected by the Avengers is part of his motivations and of the 'serious' part of the plot, which is focused on his own universe and his girlfriend. Earth-616's Avengers though? The namedropping for nerds is an absurd choice when you think about it, which the movie does not want you to do, since it makes zero jokes about it. Without that caption it was not even going to raise any objection or paradox; he could have met with the Avengers of his universe, any universe - later in the movie Wolverine says "F*** the Avengers" as if he knows them.
Two things: 1. Even if he is able to join the Avengers on Earth-616, being part of a team might give him a sense of belonging and change his whole attitude. It wasn't to impress Vanessa about being an Avenger, it was to change his attitude which had soured after the events of the second movie. 2. It cleverly explains that the Marvel characters from the Fox universe were in their own universe and, to this point, have likely not been a part of the MCU.
Maybe his universe doesn't have Avengers and he found out about them through the MCU. That's what I'm saying. It's just a gimmick in the movie and actually doesn't affect the plot that much as he was given a task by the TVA (or whatever lone wolf from the TVA) and teams up with Wolverine. Him trying to be an Avenger and talking about it is just a running gag.
Other mistake: During their first confrontation with Cassandra Nova, the dynamic duo seem to score a point when Wolverine backstabs her in a very literal sense. While it makes for a cool moment, it also is pure nonsense; they are in an open space surrounded by her goons and none of them has the slightest reaction nor you can see any cover Wolverine could have used to sneak upon her. Even the soft ground is intact.
Suggested correction: Cassandra uses her powers to phase shift (ghosting, selective intangibility) Wolverine into the ground and behind her. After her quite long time inside Deadpool's head, Wolverine manages to get himself out, phasing himself back behind her and stabs her. Since he was phasing, the ground won't be disturbed.
But he DOES disturb the ground when she phases him into it. He sinks in a hole, he leaves a trail. And that is with her controlling the process. Since he doesn't control the phasing and how to emerge from it, I don't get how he can just casually pop up (with no particular speed) without any trace, and again, unseen surrounded by goons watching from every angle, to nobody's reaction.
Look at the scene closely (if you can). There is far less disturbance than would have happened if someone was dragged through the ground. The only disturbance you see is from his claws still sticking out of the ground as he is being dragged; his body has zero effect on the ground. I think it also has to do with the amount of force Cassandra uses to pull him. Coming up slowly would hardly disturb it. The goons won't interfere; they know what she is capable of and has nothing to fear from these two.
Will of course check out the Disney+ release in the future, but the movie doesn't show the action going on this way; if they wanted to show him being phased, then they shouldn't have depicted him as being sucked into the ground with his body looking very much solid, nor his claws leaving claw-shaped trails. The fact that it causes much less of a disturbance than it could have is because well, her powers are not something we have a real life comparison with; the way she "skinned" Johnny wasn't physically accurate either but there's no lack of consistency with anything else. As for the on-screen portrayal of the ground pull, all I am pointing out is that he very much leaves physical and permanent trails on the ground that the movie shows, at no point his body shows to be immaterial, and then a minute later he just pops up, with no particular haste, and there are no traces of him going through the same medium. As for the lack of reaction, it's a lack of timely reaction; they do react to him when he stabs her, you can see some of them raising their guns, so it's not as if all of them have such trust in their boss' abilities that they are nonchalant about whatever is directed at her. It's just that they react to it when the audience does. There's no reason why they wouldn't do it earlier. Other than the fact that it's a movie, but's not like Deadpool makes a joke about their terrible reaction times.