Edinburgh Native here, about to try to get into Uni's in America. But that's a different story.
Here's some low-down on Edinburgh - my parents work at the university and rent out accommodation to postgrads, mostly in the Stockbridge area. It is a nice area there, but personally I would find it too far away from university (and it's a lot of uphill). The bus system probably seems amazing to Americans, but by UK/Europe standards it's nothing special.
Barony Street area is quite funky, and also has a tremendous butcher. (Crombies: top sausages). Some cool shops, a good fish and chip joint, lots of restaurants and shops nearby. Cowgate area is probably closest to being 'party'. But it's also dirty and drunken. Many (undergrad) students love it, but I probably wouldn't recommend it for postgrads. I generally dislike cowgate, but it does have some good bars.
Nicolson Street - you should be able to get a 2 person accomm, 5 minutes away from George square and other uni areas, for about £600 a month.
You can get stuff cheaper elsewhere (maybe £350+?) and you can certainly pay a lot more. The whole city is packed with other international students, especially postgrads, so I really wouldn't worry about credit etc. Furnished flats (apartments) is the standard, especially for students. You would be mad to ship stuff over. Most of the people my parents rent accomms to are PHD students, we've had them from Thailand, US, Canada, Japan and several from Germany.
Areas to avoid accommodation, no matter how cheap include most of Leith and Wester Hailes/Sighthill.
On the mice thing...yes. There are mice sometimes - most student houses will be between 150 and 600 years old, so there's not much you can do about mice. It is worth knowing that rats do not tolerate mice living in the same space (anywhere in the world, not just Edinburgh). So if you have mice, you don't have rats. We got rid of mice in our house by getting a cat. I think it works out cheaper than mice traps and poisons.