First off, I don't have either of the Master Index compatibility mods checked in Date Files, just the GHD one checked. Will the mod still work without those two? Cause I just keep getting error messages saying "Master Index file not found" and than something about the Warp_Andra script or something.

Also, somewhat related, are there any Console Commands to start specific quests? I want to start the script where Dagoth Gares gives you Corpus to make sure the mod works before I trudge through the MQ only to be potentially depressed that, for some reason, the mod didn't apply. Any help would be appreciated!

Everything you need to start vanilla quests is here: http://www.uesp.net/wiki/Morrowind:Morrowind

As for within the mod itself, I'm sorry, I haven't played it.

I believe it should.
We can help better if we know what plugins you are running and the exact wording of the error messages, so https://sites.google.com/site/johnmoonsugar/Home/getting-help-with-morrowind.