Did the boy intend to save Alejandro Gillick by shooting him through his jaw?

No, he did not save him on purpose. If it was written that way, then it is very bad writing.

Hitting your target exactly where you want to hit it is extremely difficult with a pistol. Firearms have recoil, which offsets where you're aiming after you fire, and the lighter the firearm, the greater the recoil. So, even at a close distance like with Miguel and Alejandro, it would be extremely difficult to shoot precisely through the jaw, especially between teeth. Treat yourself to the shooting range and fire a pistol at a basic target sheet with a human silhouette and try to shoot it between the eyes, you'll see what I mean. It's still challenging with training. A part of achieving a bullseye is using your posture, breathing control, and muscles to keep the firearm as steady as possible to prevent offsetting via recoil. Miguel was just a feeble-bodied teen with (I'm guessing) zero experience with firearms. The flimsy, one-handed way he held the pistol tells me he has zero experience. So, it's very doubtful he could pull off such a precise shot using one boyish arm and weak fundamentals.

Additionally, Miguel looked subtly disturbed with himself right after the shooting and then he jumped out of the truck not long afterwards, which tells me he really believed he killed someone and didn't want to do it again, so he was getting away from the situation that compelled him to do it. Then, there was his surprised expression seeing Alejandro again a year later. It seemed more than just an implicit, "Oh no, you again?" reaction. It seemed like he thought to himself in that moment, "How? I killed you!" Genuine surprise.

I noticed on the film's Wikipedia page that someone elaborated in the plot that Miguel meant to shoot Alejandro in the jaw. I don't know if you saw that (it has since been edited), but it's completely asinine, and remember that just about anyone can edit those pages. I personally thought the film was a mess, so I understand the confusion regarding what happened in some scenes, including what you're asking about.

Sorry for the length. I hope it was worth your time.

No, I don't think so. It seems to me, he just wanted to get over it, by shooting a person for the (supposedly) first time a milestone was achieved, then pack thing and leave, nothing else to think. It seemed pretty plausible.

Depends on the weapon honestly. In order to know if he did it on purpose or not you'd have to know if the gun recoiled before or after the bullet exited the chamber. If the weapon recoils after the bullet exit's the chamber, than it is fairly easy to place a shot where you want it. But if the weapon recoils while the bullet is exiting the chamber even the most skilled shooters will have time placing that shot even at point blank range.

In essence the prior answer isn't wrong at all, it just requires more information to determine.