1) for tar files you need a tar-program to untar them, and for gz you need gunzip

they're packages, just like in windows you had winzip and .zip -packs etc. I guess your RH already has these apps so (this is terminal-based, you might also have an application for this that's graphical...something like kpackage was back in the days?)
so, in terminal, to extract a .tar -archive, simply do (don't write the dollar-sign, $, that's here just to indicate the shell prompt)
$ tar -xf packagename.tar
and for .gz package,
$ gunzip packagename.gz
or, if you happen to gave a package with a .tar.gz simply do
$ tar -xzf packagename.tar.gz
and you have the package unarchived

then just cd into the dir, or open it by any way you just can, and use what's inside. if there are some shell scripts, you can run them like this (if you're in the dir)
$ sh shellscriptname.sh
-OR THEN-
$ ./filename.run.or.what.ever
or if it's a normal way compileable app, then usually it's just
$ ./configure
$ make
-IF YOU NEED TO BE ROOT, THEN DO THIS OTHERWISE JUMP THE NEXT 2 LINES-
$ su
$ <write the root password>
$ make install
so the "su" and password are only needed if you can't do it without
2) XMMS is quite a good one, I've liked it always. just like winamp, but better (hehe

ok, just like winamp)
3) I guess, if you need to run apps from the terminal but want to keep the apps open and close the terminal, then you might want to try using nohup ("nohup - run a command immune to hangups, with output to a non-tty")
$ nohup program_name
or just run them without terminal, like from KDE/Gnome/Fluxbox menu, icons, or something if you happen to have them

but I guess you asked for the one I told above.. (I never use it myself because I like to keep terminal window open, I just run them with & sign at the end of the command so that I can use the same terminal for many apps)
4) yes, you need an app for that, somekind of application at least. I use Mozilla's Thunderbird, and have used many others too...the "smallest" mailchecker I can remember I've used was called "fetchmail" - I got bored to sendmail-type thingies though so I gave up of it later...