I have the latter and it won't get as white as the tub no matter how much scrubbing/bleaching I do. 

However, they left some rust stains in the corner of the bathtub (what looks like from a metal can they had there), I've tried CLR, Lime away, bleach, etc. and I cannot get this to come out. 

As a Host, do you supply cleaning supplies for your cleaner (bleach, soap, detergent, etc)?

I live on a river bank so a couple gallons of bleach should give me enough drinking water.

I'd think this is prime opportunity for cleaners to charge extra and claim they're 'disinfecting' but really be doing the same thing as always; maybe switch over to bleach if they use a more natural product. 

If we eliminate the source of the mold and bleach the hell out of the place, what are the odds of the mold returning?

Splash some bleach in the cat litter, and let it sit for a day or so.  

Mold can be resuscitated simply by introducing water/moisture, unless that liquid contains an agent capable of killing the mold, such as bleach.  

I wouldnt assume there would be much mold because it is a dry environment, but possibly do a once over with bleach, fan and air it out for a few days, fresh coat of paint, and call it good.  

NOTE:  bleach/water combo will only wash away the surface discoloration.