A quicky, 'cause I gotta run to work.
First, the RIAA is not an "artist rights organization." It's the recording industry group.
The others are performance rights groups. As a performer, you basically don't have to deal with these groups. That is the responsibility of the venue. The venues (concert halls, bars, restaurants, etc.) generally pay set fees to the groups, allowing people (you) to perform covers of tunes by artists registered with them.
For mechanical and digital rights, you need to either go to Harry Fox Agency, or to the publisher, or in some cases to the artist. It all depends on who is handling these rights. "Big names" generally contract with HFA or someone to manage their mechanical rights. Smaller artists either let their publisher do it, or handle it themselves. (Lots of people just write to the artist and get permission.) It takes a little research for each tune.
What you encountered with multiple listings and listings of PD songs is probably arrangements. (Also, of course, actual recordings, which are of course protected as such.) That doesn't take a PD song out of the PD.
As for figuring out whether a song is public domain, yes, there can be a certain amount of guesswork involved. Some songs are known to be PD ("Soldier's Joy," for example), while others may be sort of in a gray area.
My own approach (as a lawyer with some IP knowledge) is to focus largely on known public domain material, and to try writing original material. Can't say I've accomplished much, but as a philosophical matter, that is my approach. Beyond that, there are some artists I feel confident I can contact and communicate directly with. And then there are the clear-cut songs that are registered with HFA. Gray area, well, I reckon you take your chances.
I think most "violations" go unnoticed, fwiw. And those that do get noticed, most of the time there is a letter from a lawyer requesting you to correct the matter, which would be easy enough for an Internet publication of a tune, but with a CD would require stopping distribution, and possibly paying royalties for those already distributed.
J. Mark Lane
Stanley #10 F5
Pomeroy #72 F4
Brian Dean #30 Bowlback