Just out of curiosity, do radio stations have to pay a royalty to
> record companies when they play music?
No & Yes -- There are three major "performance rights" agencies in the US. ASCAP, BMI and SESAC. These organizations collect royalties form radio stations and distribute them to the composer, lyricist and performer, but not to the record company.
For reasons that escape me right now, but probably having to do with radio being a giant advertising medium for the record companies, no royalty is paid to the record company.
What are the rules, if any, for someone recording a song off the
> radio or a tape off of TV?
Depends on who's point of view is being presented. As long as its for personal use, most legal authorities would hold no foul. Hand the copy to a friend? Technically wrong, but you aren't likely to be tagged. Sell the copies on eBay? You'll get tagged quickly.