What is the reason for not giving SSN? Anything is possible but I will think ten times before renting to a person who doesn't want to disclose required information. I used to trust people and rented my properties on parent's co-signatures but in EACH case they trashed the apartments and parents could not come up with rent even. You can take them to court (most of the time out of state) and get the judgement but can you collect the judgement? Is it worth it?
It's better to get a good, hard working tenant who cares about his/her credit score and lower the rent or give last month free.... more
You can get a mortgage, but there are only a couple of banks where you can get such a loan (if you pay taxes here and can show a lengthy work history, you may be fine with other banks as well).
As far as the interest rates, most of those loans for foreign nationals are based on the LIBOR (the British version of the US Federal Reserve Rate) so rates will vary from typical loans you see advertised. As for how much different they are, you would have to speak directly to the bank about rates (and eligibility).