syncs right now go from free to maybe 50k, unless it is a massive song (thing the Beatles) in a massive campaign or feature. the average network tv sync right now is prob around 5 grand all-in, meaning 2.5k goes to the owner of the sound recording copyright (the label or artist) and 2.5k goes to the owner of the composition (the songwriter(s) or publisher(s)).
Music publishers can take anywhere from 10% (for an admin deal) to 50% (for a co-publishing deal). Bigger percentages involve advances (recoupable payments against future royalties), but also much longer terms (5-10+ years).
Songwriters don't get publishing deals unless their songs are being performed or sold, and there are lots and lots of indie publishers out there, along with the majors as you mention.
In many territories outside of the US, artists and master recording copyright owners do get paid for public performance - in the UK for example PPL is one society that pays some of these rightsholders for public performance.