Is there a way to get paintings into Nook's catalog in City Folk like you can in Wild World? If so, how?

Buy them.

I assume from Nook himself. I bought one from Crazy Redd that I already have in the museum. I sold it to Nook and thought that it would end up in the catalog. That should've worked, right? If they actually appear in Nook's store, when?

They very, VERY rarely appear in nooks store. Also, I don't believe you can order paintings from the catalog in ww. However, they should appear in your catalog once the item is in your pockets at any time. If not, then you may not be looking in the right place or it may be a glitch.