My right hand 9-speed 105 brifter is acting strange: when I brake, the lever doesn't return all the way back, but get stuck a little bit before (as if I had an invisible spacer there.) Pushing the lever a tiny bit to the inside (as if initiating a shift to a bigger cog,) makes it pop into place.

To my untrained eye, it looks as if the shifting/right-left spring is misaligned, and thus pops back a bit to the left then it should.

I know that Shimano levers are mostly non-serviceable, but I'm hoping this part is. Is it?

Try cleaning or replacing your brake cables.

Disconnect your brake cable. Does it return ok now? Then it ain't your brifter.
Compress your brake manually. Does it return ok now? Then it ain't your brake.
Next step, as previously noted, is replacement of cable and housing.
In my experience wrenching a brifter will go teets up on the shifting long before the brakeing.

The cables are quite new (installed about a month ago,) but I recently got a new handlebar tap put on (by what seemed to be the dumbest employee in the store,) so it might be the under the tape housing. Could this be the culprit?

My bet is pinched cable housing under the tape.

Yep ! Most likely.

Aslo, if you ride in bad wet weather all the time it could be the cables anyway.

Do exactly what capwater says.

Rant time, please permit me to vent. I apolgize in advance if my brashness offends you.

1) If you pay people to put on new bar tape, then probably debugging and fixing your brake problem is beyond your level of mechanical expertise.
2) Why do people who just paid to get a bike worked on (by an obvious dumb employee) seek advice on this forum on how to fix someone else's f*ck up? You just spent money, something isn't right so take it back and tell the shop to make it right.

True, but then the other side of the coin, is figuring out what went wrong and the knowledge you gain from making it right. And you can do this without even leaving your house.

I didn't. I admit that my mechanical expertise is limited, but putting a bar tape on is something I'm perfectly capable of. Actually, I really like putting tape -- it's oddly satisfying.

However, I had a nasty crash the other day, so I rolled to the nearest shop I found and remembered hearing nice things about, and asked them to take a look at the bike, just to make sure everything is indeed unbroken, and that I won't kill myself on the way home. I left the bike there and went for a coffee, and when I got back, they already put a new tape (the one I had got ripped in the crash, but I planned replacing it at a later time)

I have no idea who's **** up is that. It could be that shop fault, which is 100km away, and I'm not going to drive to just to yell at them, when maybe someone else is to blame. I'll roll the bike into my LBS tomorrow, but if it's something I can fix myself today and still get an early spin tomorrow morning before the shop opens, all the better...

Bummer about the crash. Check the cables, most likely that's it. Ride on .....