AF and AE are separate.
You can turn autofocus off and on (and lot more options besides) totally independent from the exposure mode.

Yes AE has two modes, Tv or shutter priority and Av or aperture priority.
Depending on what you're trying to do you will want to vary aperture or shutter speed and let the camera work out the other one to get the right exposure.

I'm surprised to see the mkIII has P mode, I'd never use it. I prefer Tv, Av, or Manual.