Partial connection drop if idle for ~5 minutes

I use a program to connect to several games. It launches a DOS window. While I'm in that window and it's active, in the foreground, I have a command set to ping the connections every five minutes or so to keep them alive. The problem is when I leave that window to do something else, even if it's only for a few minutes. I come back to the DOS window and all my connections have been dropped.

However, this does /not/ mean that my entire internet connection has dropped. Just those specific ones. I can still have a Putty window open the entire time.

1. The DOS program sits idle and inactive when I switch to something else. Is there a way I can keep it active/updating and still browse the web, etc when I want to?

2. Is there some setting on my computer, or my card, or something that I can set to make the idle timeout longer... or whatever the problem might be there?

3. Is there some setting on the router (or the modem) that can be tweaked to fix this?

I don't have this problem with all ISPs and router setups, which leads me to think it's either the ISP or the router. However, there may be something I can do from my computer to fix it as well, or work up a workaround.

Details: I have a Dell Inspiron 1150 laptop with a Dell Wireless 1350 card. I'm running Windows XP. The router is Netgear. I believe the ISP is Covad DSL. The program I'm using is Tinyfugue TF 4.0 alpha 4, ported for Windows.

For those who know what Tinyfugue and MUSHes are, my problem is that unless each world is pinged in some manner (such as /repeat -00:05:00 99999 /send -W @@ Ping!), then if it sits idle for more than about five minutes, it drops the connection. TinyMUSH doesn't have a ping command like some games do, so I can't set it from the game end.

I'd appreciate any help or suggestions people have.