Hi srlamb! Thanks for your response and your help.
1. As I found now, the host in question ("www.shlomifish.org") does not answer to pings even if I ssh there and ping localhost.
2. Interesting. I wonder what can cause that. I should note that we replaced a NAT/router box around the time these problems started to appear. Maybe this box is buggy in this respect. One thing I can do is make a packet capture of the nearby Windows computer for a "lynx http://www.shlomifish.org/" command after it is broken on my Linux computer.
7. I see.
Another thing I noticed yesterday was that after the problem appeared I still had an ssh connection open, and it worked perfectly ("ls -lR" on my home directory there and all). After I disconnected it, I was able to reconnect again (after one failed attempt), but not more.