They have a blanket licence agreements with PRS and similar organisations. IIRC is only covers using some number of seconds of any given track, and to use an entire piece (or larger part there-of) separate permission must be requested (and perhaps paid for).
Yes, the BBC used to license tons of music (still does?), they had some really cool background music for all sorts of stuff by really good artists. Late shows on Radio one used to be legendary for the variety of alternative music they played
Yes they can, all they need to do is stump up for the licenses the same as the BBC do via organisations such as the PRS:
https://www.prsformusic.com/
https://www.prsformusic.com/licences/broadcasting-music-on-t...