I'm confused about this because technically bumping threads can violate the double-posting rule. Can someone enlighten me on this?

Bumping means that your thread didn't get attention so you just post uneeded spam into it to bump it up in the ranks so people will see it more.
Double posting means posting twice with exact messages.
They're different things, but that still doesn't mean you should do them.

It depends. If your making one post right after the other then that is considered double posting. If you are doing it to revive a thread that is fine, but I say it should be at least three/four weeks old. And obviously the thread should still be relevant (so don't go bumping something like "Super Mario Galaxy 2 announced!" or "Are you going to get HeartGold/SoulSilver"?)

So if I wanted to show everyone a new picture I made and my art thread is old, can I post that I have a new picture with a description?

Pretty much (although this is an assumption), you can bump threads if these apply.
1) The thread still has a valid purpose
2) Your post has a valid purpose
3) The last post on that thread wasn't by you.

You forgot to say that whenever a thread is sticky, you can post in it whenever you want.