You won't find apartment complexes like in the US, they will generally be single flats or houses that are privately rented by estate agents. You will need at least 1 month's deposit, and depending on the estate agent/landlord, you might need a pet deposit and or 6 months in advance. I find Zoopla.co.uk is quite good for finding flats, and it lets you choose the option of 'pets allowed' in your search. It depends on where you'll be working, but I would suggest places like Musselburgh, Portobello, or Dalkeith if you'll be east, Penicuik if you're central, Currie or Balerno if you're west, or South Queensferry/Dalmeny if you're north. All have multiple direct bus and/or train links into Edinburgh.