Amusingly enough a male Mary Sue is called a Marty Stu or Gary Stu. Your definition is a bit off as well, usually the key aspect of this type of character is that it is an author insert. So they are overpowered and overrespected usually because this is what the author wants to be. However this is a tiny nitpick compared to your overall point.

Finn has the backstory of being taken ... [...] ... deciding to go with his dick and save Rey.

They specifically mention that they have problems with their recruits and that they might have to get clones. The raid was also Finn's first sortie, he reveals that he used to be some sort of janitor or maintenance guy something. I agree that he is too teenager like.

Plot needs someone to rescue Poe.

I think Poe is the device here. He barely did anything and is billed as a secondary character.

Realistically how would he actually be able to figure out where the base is weak as a newbie Stormtrooper anyways?

They made him a maintenance guy so that he would have knowledge for the plot.

Hollywood needs someone for Rey to have romantic tension with

Maybe I'm wrong but everything seemed to indicate that he has been friendzoned.

Despite Finn's background everyone just instantly trusts him.

He is characterized as a little rogueish and a good liar. As is stands Solo was able to see right through him because he has the same qualities in droves.

Despite Finn's lack of experience he is much more skilled at fighting than most of the other Stormtroopers (and the guy who beat him down didn't even pose any real threat) Despite his lack of skill Finn managers to go and undermine Phasma (the silver Stormtrooper) as a credible villain.

Good point

to be loveable with unwarranted skill and no consequences to his backstory.

He is portrayed like a bumbling baby though.