Shipping your car will be a very expensive proposition, if you have to pay it yourself. Better buy brand new here since QP will give you interest free car loan. Check the classified section on this site or go to Qatarsale.com to have an idea of what brands/models are available and their prices. Most american i know go for big 4x4 SUVs as they don't need to worry about petrol price....

QP should provide you with temporary accommodation (a compound villa for family) for up to 3 months (?). The house hunting during that period should not pose a problem as there are many accommodation options available nowadays to suit every taste and budgets.

The biggest worry, as other had mentioned, will be getting shools for your kids. ASD, American School in Doha do give priority for US passport holders, but as far as I know it is not guaranteed.