Okay, so I'm not exactly clear on whether or not you can have both Clonewars and Warlords mods installed at the same time, and if so, is there anything special that you have to do to make them work?

Yes, you can have both. They run on different shortcuts

And there isn't anything special that I have to do.

Absolutely nothing else, the installers make the shortcuts with teh correct settings, so just run those, as Evaders99 said.

Thank you. And nice avatar.

Okay, so I installed Clonewars and ran it for the first time. I picked the Republic to play as, and started the game. All the Clone Wars style graphics of seperatist ships were there while the game loaded.

 

The game starts, and a Hiigaran Mothership and carrier are sitting in front of me. I build a fighter facility, and then an Interceptor squad. The ships are ALL HIIGARAN. What gives?

It is not designed to be a single player mod.

I WASN'T playing on single player. I was playing skirmish vs cpu with skirmish as the gametype, and when I tried it again with the seperatists with Clonewars .9 as the gametype, it worked.

Well that is probably your problem, I personally have not tried Clonewars with the Skirmish gametype, so i don't know what heppens there. But since the mod is Clonewars, i would say you should use the clonewars gametype, for a error-free (somewhat) experience.