If me and my friend want to play one of my old maps with EUD triggers, is there any way we could both download an older version of SC and play it together on some special server or something?

The same goes for mods. Can it be done?

Yes. You can download a downgrader to do it for you. Here's a good one that's updated to the most recent patch:

http://www.broodwarai.com/forums/index.php?showtopic=374

EDIT: If that doesn't have the patch you need, try:

http://farty1billion.dyndns.org/SCD3/

Well, I already have the downgrader, but my question was whether we could play together on the old version somehow. I once saw a program used for playing mods online but I can't find anything like that now.

There are certain servers that let you play with older patches, you can use those.

http://www.staredit.net/?p=modnight

Check out the 'setting up your computer' section. It'll teach you to put a private server into your registry that you can play on with your friends on any version.