1- You need a Qatari guy who will hold minimum 51% of company shares (by law). he will have full control over your life. ie, u need his permission to leave the country, issue a driver's license, buy a car, ...)
Then you need to prove that you have a place for the company (electricity bill, Banner)
It will take you around 2 weeks to finish all the paper works needed to get registered and have a company number.

As for openning a US dollar saving account, as far as I know you can't until you have qatari residence. But you can open a Qatar Reyal saving accounts.