One of the reasons for not getting paranoid immediately is that there are enough things that could cause just about any "strangeness" that you can get yourself confused if you rush into trying to fix everything that might be wrong all at once.

I don't recognize the "domains by proxy" that Newport Boy mentioned, and first thought was it might be one those "redirector" services that let you send email without showing the real address of the sender. Maybe someone knows more about that(?). This would suggest that whoever is sending them might be working a little harder at it.

Any recent Windows versions (prior to Win8) lets you find the Command Prompt at Start|Programs|All Programs|Accessories, if memory serves me. Vague memories say it might be a step further, at System Tools or something similar in some older versions(?) but it shouldn't be hard to find. You can also get there if you hit Start and put "Command" into the "search box" at the bottom and hit Enter.

"ping mudcat.org" in a Command window would send a ping to mudcat and tell you if it got through. One of the handy things is that you can use the "name url" (e.g. mudcat.org") or the "numeric" one, but the ping will report back using the numeric address (173.163.150.105 for mudcat), which is the simplest way I've found to find the numeric one when all you've got is the "familiar one."

John