Also, I purchased trinity. Is there any way to play regular sins online with a friend who has the plain version, or does he need to upgrade to diplomacy?

When you start Sins of a solar empire, entrenchment or diplomacy.  Just start the sins of a solar empire if thats all your friend has.  However its strongly recommended he upgrade to take advantage of all additions to the game ie starbases in entrenchment and extra ships.

I see where I was confused. When I launch Sins from impulse, it goes straight to diplomacy, but I've figured that out now, but I'm still having trouble downloading updates through impulse. Is there somewhere I can get them from somewhere else and install them myself?

Before you say you have problems updating, actually check your game version with the most recent version.
(You can see what the most recent version is in Impulse, I believe)

There seems to be a LOT of confusion over "do I have the latest version" because all the expansion packs have different version numbers!

A good start would be to completely uninstall sins of a solar empire and then reinstall trinity via impulse if one of your versions is back a bit.