After I run DF on DOSbox, it keeps the DOSbox cursor instead of switching over to the DF cursor, so I can't load a game or start a new one, just aimlessly click at them.

Press Ctrl-F10.

Well that was simple.

Another issue, and I have this with Arena too, but when I go to pick a province, I can't pick the Imperial City. I click everyone on Cyrodiil, but nothing happens, where as soon as I click another province, it moves on. Any way I can become an Imperial with this problem happening?

You can't be an imperial.

Specifically because there wasn't an "imperial" race before Morrowind. Cyrodiil was just a melting pot of all the other races.

"Imperial" is a very ugly name for a race.

Oh, I did not know that. Interesting, but thanks guys for the help.