Already, let me chime in on this OP, and that other guy with a type-S.

Had a 02 Base 5-speed for my first car. Few years down the road after starting it would make a little grinding noise which I hated, but couldnt do anything about it at the time. Eventually sometimes it would take me 15 minutes to start it, so at that point I ordered a new starter. In fact, I got the same exact one as you from autozone.

First thing you shoulda done differently is you should have done the install yourself. I don't know how handy you are with cars, but it took me and my dad about 2-3 hours no stress to replace it. The coolant lines drip a little bit, but it doesnt pour out. Its an extremly basic repair IMO. As for the intake gasket, I never bought a new one. Used the old one (look fine to me, no wear or tears) and didn't have any problems.

One problem I did run into after the install was my idle was fucked up. So after some searching I removed my IACV and soaked it in cleaner/cleaned my throttle body and everything went back to normal.

It sounds like the shop fucked your install up somehow, but I can't really imagine how. They might have moved things around in your bay for room/clearance as its a pretty tight bay to work in and possibly moved your CAI to far in/close? I know you need to remove the air filter/CAI for this install. And if you need to remove the intake manifold again, I doubt you need a new gasket, but I could be wrong on that.

To the other member who has a grind on cold starts, it has something to do with the K series engines. Apparently theres a petition (lol) for the K24 to get a recall on it, but Acura wouldnt even recall DC5's for the 2nd gear grind, so I can't see them doing anything about this. My DC5-S has the same grind, as did my base. Usually its around 35 degrees or lower. But yeah its not hurting the car, it just sucks lol.