For some of the more recent music in the games like the Sinatra stuff and others, how did you get a hold of the licenses for those?

You buy the rights to the song, or you pay for the use of them.

I also have a few more questions about this same topic, kind of (if your willing to answer).
1. Are any of the wasteland radio songs on the public domain? (which ones?)
2. If I were to make a fallout film and/ or animation, how would I go about getting permission to use the score written specifically for fallout if possible?
3. Would I be able to monetize said video (only on youtube, deviantart has no monitize option)?
Thank you if you answer these questions.

That I'm not sure... I would recommend looking at the songs on something like WIkipedia to locate one of the companies who released the song you're after on a record, and working from there.
You need to contact the license holder, or their agent, directly
That depends on the license you negotiate.

I used to work in a radio station, and never mess with music licenses, most music like most films cant be played or displayed in public unless you pay a fee, theres alot of the old songs in fallout that are public domain, i did look some of them up, because i was wondering when fallout new vegas came out, i thought there would have been alot more music, but there wasnt and i can see them not wanting to pay a fortune on licenses.

Back in the 90's the music companies really cracked down on where music was being played because alot of shops were playing radios, and gyms had aerobics classes where they were playing popular music and all those places werent paying license fee's, needless to say alot stopped or found public domain music or free music from unknowns, some payed license fee's, and alot got taken to court.