Name of this part.
 

Sorry, I don't know all the technical terms but I am painting my bike frame myself and this piece seems like it wouldn't really be worth painting. It is the part on the chainstay for the real derailleur. It is detachable. I can't get the black plastic piece out without destroying it. I wouldn't mind if I could find a metal or plain black piece like this. Or maybe if I could simply replace the black plastic piece after I destroy it, then I wouldn't mind painting this piece along with the frame. Any help? Thanks.

Looks like a CABLE GUIDE. The black plastic tubing is nothing, u don't need it. The brown block is probly attached to the frame with a screw, of which u will see once u remove the plastic tubing. No guarantee. Even if the plastic piece is riveted, u should be able to drill it out, tap it, and replace it with a standard screw.

PEACE!

Thanks. The screw is actually on the other side which is not pictures. I have already removed it but I put it back on just to take the photo. If I do not need the black tubing piece then I will just remove it and paint the rest of this part along with the frame.

Black plastic tubing piece is called a "ferrule", kind of like the white plastic tubing piece at the end of cue sticks for pool/billiards.

The black plastic tubing is a FERRULE. You DO need a ferrule, especially on your read derailleur cable. You can get new ones at any bike shop for very very cheap. The ferrule makes the end of your cable stonrger and gives you better shifting. If you decide to go without a ferrule your cable will eventually peel apart and make you shifting stop working.

Go to a bike shop with your cable housing and aske for a ferrule (or cable housing end) that fits your cable. A ferrule that doesn't quite fit is better than no ferule at all.