What happened to the Satellite MP3 download program (you can no longer
download music).  Are there any other widely-used software/services
that allow the downloading/sharing of MP3s?  Are there any overseas
online services that are immune to the laws/lawsuits of the US?

The satellite program you refer to is the AudioGalaxy Satellite.  This
quote and following article will explain the current status of their
operation:

"Audiogalaxy reached a legal settlement with the recording industry
Monday, agreeing to obtain permission before allowing people to swap
copyrighted songs through its popular file-trading service.
  The agreement halted song trades on the service Monday night.
Searches conducted by CNET News.com for tunes by musicians including
Lauryn Hill, Marvin Gaye and James Brown returned no results or an
error message saying "search prohibited."
http://news.com.com/2100-1023-936932.html

As for other programs go, there are lots available.  Here are the top
programs sorted by the amount of downloads last week:
http://download.com.com/3150-2166-0-1-5.html?

Your third question is a great deal more difficult to answer, but the
short answer is simply "no".  I'm not really going to get into all the
legal mumbo-jumbo, but it's really difficult to have a large
underground illegal (which is what it would be) network like that. 
For example, here's a recent case about a movie piracy website in
Iran:
http://news.com.com/2100-1023-942856.html

As I'm sure you can imagine, we (the US), don't have the best of
relations with Iran at this moment, but the website was still shut
down.

There are three basic prinicples to this.  Number #1, anything illegal
like that doesn't make a lot of money and the world is driven by
money.  Number #2, anything illegal is illegal and therefore unlikely
to become large because most people don't like illegal stuff.  Number
#3, unless it's large, a file sharing network is pretty worthless.

That's basically it.  One way or another, a large underground network
wouldn't last very long and it has to be large or it's of no real use.
 Just stick with the other programs in the download list and you'll be
fine.  There's really nothing that's "immune" to lawsuits.  They'll
sue whoever they have to stop it.

Hope this helps you.  Please feel free to ask for clarification if
necessary and if you're satified with the answer, please rate it.
Thanks!
   
xemion-ga

Wonderfully helpful.  Saved me tons of time, thank you.

A service you might want to try is http://www.mp3raid.com/ . However,
in using this service, please be sure you have the rights to download
any song which you find.