I was wondering.........

should the LimeWire app close when I close the terminal window that I used to open it with?

yes

yeah that's supposed to happen. you might want to check out some of the posts about making a (desktop) shortcut

I feel it is best to stay on the lime wire network so this is how I keep my limewire running. I start a vcn session
vncserver :69 -geometry 800x600 -depth 24 -name limewire
Then i bind to that vnc process
vncclient my.domain:69
Then I start limewire from within that vnc session. then i can close the vnc viewer and limewire keeps running.... This also allows me to use limewire at work becasue I can still connect to my machine to the vcnserver process from work through a socks proxie... Just an thought.....
Where there is a will there is a way!!
MC

If you start LimeWire from a terminal with, say, ./runlime.sh then LimeWire will close when you close the terminal.

The way to get around this is to background the runlime.sh process so you can close the terminal without terminating the runlime.sh process as well. To background the process start LimeWire like this:

./runlime.sh &