Dreamer13, most of the things that you want to do here in Qatar relies on ur RP... so be sure you have one so as to do all the things you want to do...

in regards with the process of ur RP, having a US passport doenst change anything... it all depends on the company if they are really interested to hire, they can do things fast...

all you can do now is inform them the expiration of ur visa so that can do something about it the soonest...

hope this can help you... good luck!!!