Looks like standard police issue, so definitely 9mm. And the 9mm isn't that big. The projectile itself is about a fingernail. Then you have it flattened after hitting Logan's metal noggin. I'm not saying it's probably or realistic or anything, but I wouldn't discount it as a possibility. A very very very unlikely one at that. Lining up of perfect conditions comic book logic etc etc.
I chalk it up to the usual "film makers don't know how firearms, and well, physics works" issue. Yes, in theory, if you had a skull made of (ok. "laced with" ) an indestructible metal, and were hit by a bullet, the bullet should either ricochet off, leaving a small flesh wound (which he should heal in moments given other scenes of him healing in the films), or perhaps, if it hits just right, at a perfectly straight angle, just deform flat and sit there, stuck to the side of the skull. We could even accept that the action of his flesh healing might push the flattened bullet out.
In the scene, the bullet that pops out looks pretty much like an undamaged bullet, clearly much longer than the presumed thickness of the flesh around that part of the skull. But that's probably a combination of the film makers not thinking about the bullet deforming, and the more reasonable need to make sure the audience recognizes it as a bullet. So show something that looks like a normal bullet being pushed out of the side of his head via some cool cgi effect, and everyone is happy. Yay!
Why it knocks him out at all or takes so long to heal from it I'm at a loss.
Needs of the plot. The primary point of that scene was to show/lampshade Pyro's shift towards villainy, so that his decision to join Magneto later in the film makes sense. If Logan isn't knocked out, then it becomes a scene about Logan beating up some cops (presumably with his fists and not his claws), while they fruitlessly shoot at him. Given that we'd just seen a scene with him slicing through a bunch of far more heavily armed mercs, it would not work (same basic fight, absent claws and machine guns). Knock him out and you get to have a different action scene with fire and exploding cars. They could have done that same stuff with Logan up and fighting as well, but it wouldn't have had the same story impact as having Pyro doing it all by himself.
Of course, having Logan up and fighting the cops in melee while Pyro blasted at cop cars and otherwise kept the rest of the folks at bay (with Iceman using his powers to prevent accidental deaths, just as he did in the film), would have actually made more sense, since it would explain away the other questionable bit in the scene (why didn't the cops just shoot Pyro?). But again, the needs of the character development outweighed the need of having a sensible scene in the first place. Sadly, there's far too much of this in modern film making.
As to the whole adamantium bullet issue (and bullets in general). I can actually see Logan getting knocked out, since having fast healing doesn't prevent concussion from what would be a very significant blow to the head. We saw Logan get knocked out in the first film when he got whacked with a tree by Sabertooth and knocked into the front of his truck. So the idea that he could be knocked out by a blow to the head is there. The problem comes in with the adamantium bullet though.
In Origins: Wolverine, he's shot in the head with an adamantium bullet by Striker. This is supposedly what causes his amnesia, and is the setup for his condition in the previous films. This raises a ton of questions though. If the adamantium bullet can't penetrate his adamantium skull, then it should do no more damage to him than a regular bullet of the same mass fired by the same gun. There's nothing special at all about the bullet in this case. So the fact that it causes him such brain trauma is presumably because it does penetrate his skull. But then the question is: What happened to the hole in his skull? The adamantium doesn't heal. It's just there. He should have a rather large hole in the front of his skull that can't be healed, and isn't protecting him in any way. Ok. Maybe his regular non-laced skull heals in place and covers the hole, but still (and let's just ignore that skulls don't just open up in neat round holes when something penetrates them). You'd think someone might have noticed that, perhaps when they were doing a scan of him, in the X Mansion, in the first film, and were specifically looking at his abilities, his bones, and his loss of memory. They had a whole scene where they talked about him, including his memory loss. A big gaping part of his skull missing adamantium might have been a good starting point.
Yeah, I get that they were filmed in different order, yadda yadda. Got it. Um... But then why not come up with something a bit less continuity breaking in the later film? They could have come up with almost anything to explain his memory loss. Instead what we got was... an adamantium bullet.
And then, as though they couldn't think of anything new and original, they bring in the same thing again in Logan. Cause... it's a bullet. And it's made out of adamantium! Checkov's bullet! Yay...
To be fair though, at least these are all just side issues, and not core plot elements themselves. In the grand scheme of things that annoy me about films, minor fails with regards to physics are well down the list most of the time. It's not remotely at the same level of Star Trek 7s "Why can't he just fly into the ribbon in a ship? You can't get there in a ship" bit, where every single character ever shown to have entered the nexus prior to that point in the film did so on a ship including the antagonist himself
. Yeah. That's still numero uno for me, since literally the entire motivation for the antagonist, everything he does, and the fact that he's an antagonist at all instead of just "crazy guy who bought an old shuttle, flew off mumbling something about returning somewhere, and was never seen again" rests on that massive gaping plot hole.
So yeah, while it bugs me when I see stuff like this, it's unfortunately something you just have to ignore and move on from. I've accepted that very very few films, especially sci fi films, will manage to avoid these kinds of things. There's just too many people involved in those projects who think that once you put the word "fiction" after the word "science", that's a license to do anything you want and just hand wave it away, no matter how nonsensical it is.