I doubt that too; maybe it's just me but it puts me off to see carpet in a bathroom situation and in a place that may or may not have moisture issues - I'd just as soon see the floor painted and put down an area rug. 

If the blower wheel is plugged up full of hair, get an HVAC tech out to clean it.Go around the house to all of the air vents, and make sure the flaps in them are open (use the little handle/lever/knob that sticks out of the grille), and that they don't have a rug or furniture or something blocking them off.Sometimes people will turn the thermostat down and then close the vents to try to regulate the temperature - if this goes on long enough, the evaporator (A-coil, inside coil) can get ice on it, which really blocks the air flow. 

You can pull the rug from under them.But I normally give a 2 month heads up to tenants as a courtesy if I'm gonna jack up the rent. 

She liked to be on her barefeet in the meantime, ceramic/porcelain seems not pleasant (rugs on ceramic might slip?).

If he does apply, you might want to see if you need to remove throw rugs bath mats, etc. 

Easy to sweep or mop, looks nice and shiny with minimal effort, and can be covered with a rug if someone wants to.

You can also suggest they buy a large area rug that they can put over top of the existing carpet in the living room where their son is playing on the floor.

They all were kinda ugly and had rugs which would require constant cleaning or replacement:Here is an after, with vinyl plank flooring: It looks a lot better, but this particular unit had a pretty even floor.   

Buy them a neutral rug for their kitchen and some plastic dots (not sure of their proper name) that go on the inside of their cabinets so they don't slam.

Particularly if you're still young without any rug-rats to worry about.