1. You can get a nice 2-3BR/2Bath apartment for about 2,500RMB/Mo. Your school or employer should provide you with an apartment, or at the very least help you look for one. Best advice is to work close to your place of employment. Typically, everything you'll need will be within walking or biking distance.
2. Nothing is extremely costly. But, you'll probably want a mattress, bed sheets, pillows, kitchenware (i.e. pots, pans, silverware), cookware (i.e. microwave, oven, dish rack), and other home amenities (i.e. internet, cell phone). Be weary of getting a 'furnished apartment' as they are very, very, basic. Its highly recommend you ask for pictures, or see the places yourself before you get one.
3. It should be one pet per passport, and it should be on a Z-Visa. I'm not sure your 'Home Visit Permit' would work. Your best bet would be to contact an animal import service directly. It's going to be relatively expensive though, a minimum of a few hundred dollars for one animal, and probably a thousand or more for multiple animals.
I wouldn't recommend brining the cats.