-If you DIY, you will have NO problem with your budget (I did mine for 2k but kept my stock HU). If paying for install, you'll probably use it up.
-If you don't cut open the doors (or the kicks), and mount the mids to baffles, you will be disappointed with the resulting mid-bass. Lot's of threads on this here.
-Lots of options for subs, music type and SPL targets needed to properly recommend. I'm happy with a couple of 8's running IB in the rear deck (clean, maintains trunk, and low cost). Others prefer more. I heard a Morel Ultimo12 with ~600W that pumped beyond what the 6-1/2s in the front stage could output.
-Clean AND super loud is relative. If you are looking for competition level, then you need to be looking at a more complex install with big mids (8s or 10s) and big amps to produce "clean and loud" air movement. My system is not that, but it gets louder than I like to listen to and it stays cleaner than most can discern, with 180w/ch on 165mm (6-1/2) mids.
-Fits together? "Technically compatible" is straight forward. The large majority of mobile audio equipment is 12V and designed for use 4ohm drivers.
The tricky part is the personal choice part of it. This is a blend of your ears, music preferences and vanity
Tweeters in particular have options...silk, metal, ribbon, ring, etc.
I highly recommend you demo some in-car systems to find out what type of sound your ears like.
From there, search "e46" in this forum, ICE forum over on m3forum, & the Build log forum over on DIYMA (where you will find my complete build log
where I included a lot of links I used for inspiration
Good luck and start a build log!