We had two chinchillas briefly when I was in high school. I'll reply to your questions as best I can using my memory alone - you can look up info on the web.
They slept in the bedroom I shared with my sister. They are nocturnal and can be quite disturbing if you are a light sleeper (I am not). They need to be let out daily, preferably more than once. We closed the bedroom door and let them run around our room. Of course they poop on the floor while out. Another thing they do is chew wires and cables and furniture. They are rodents and need to chew to keep their teeth a manageable length. So, they should have something to chew on in the cage.
They come from fairly arid climates where temperatures can fluctuate a lot. I think that cold temperatures would be harder on them than hot temps. You should be aware that they clean themselves by taking dust baths and need special dust to roll in (can be a little messy and likely not so good for your lungs!).
To answer your questions:
* I think you should get two, but the website I linked to below seems to think it depends on the nature of the individual animal.
* How many days do you spend at each house? Could you bring them with you and have two cages at each house?
* I have no advice as to where you should buy them.
* They get to be about the size of two adult hands cupped together.
* The mountain regions of South America can be both hot and cold, I think they can handle no AC - where do you live?
* I would keep the chinchillas separate from the cat when unsupervised!
* As noted above they are nocturnal. I'm not sure if you can train them out of the habit of being up at night.
* They are very soft and sweet and fun to watch, but I personally don't think they'd be a good replacement for a cat. Cats are much more interactive with humans.
http://members.aol.com/sirchin/faqs.htm