What is/was Broadcast?

Andrew Lenahan's Broadcast is a half-hour radio comedy/variety show conceived, written, produced, and for the most part performed by Andrew Lenahan.  It ran from 1998 to 1999 and is currently defunct.

Will there ever be a "Broadcast" best-of collection, or a box set of every episode, or streaming media/MP3 of the full episodes or clips?

The material from Broadcast will eventually be released in one form or another.  individual CDs of all the episodes are probably the most likely to happen, and would hopefully please just about everyone.  A box set (with some sort of special packaging) is another distinct possibility.  A best-of sort of thing is possible, but towards the less-likely end of the spectrum.  MP3 and streaming media are also unlikely because the sound quality would suffer.

What were he two different theme songs?

The first episode had no theme song.  In the second episode, the theme song became the traditional "The Wheels On The Bus" with lyrics that often changed from episode to episode.  "The Wheels On The Bus" was chosen because it's probably in the public domain and we didn't want to have to give anybody actual money.  Later on, an original song was used, "Broadcast Theme".