If the port 6436 test doesn't work, it can point to interference from an ISP.

Some people have had trouble with Charter limiting download speed & also blocking ports. A Google search should bring up some info

Just check with whoever's in charge of the appartment's internet & make sure that they haven't made any changes that could block ports.

Try changing ports first. In LW go tools > options > advanced > firewall. Change LW's listening port to 65000 (you can also try other ports in the range of 49152 - 65535) > apply > ok. Restart LW & see whether it connects.

If that doesn't work, delete LW's preferences folder. There are instructions for that & other fixes to try within this thread: If LW Will Not Connect

If you're using p2p apps (like LW) it's a really good idea to have some antivirus protection... there are so many infected files being shared. Even if you're an experienced user, there are new threats popping up all the time. My advice would be to find a good AV & use it to scan every downloaded file, before you open it.
This post has links to some free security programs, just in case you're interested: http://www.gnutellaforums.com/330543-post11.html