It would be Abu Dhabi where most of the consulates are located.

It really depends. Many embassies in this part of the world require you to have a residence visa (not just a visit) for the country in which you are applying for a visa. It seems to get less open each year for whatever reason. For example, India: in Qatar, they won't issue a visa to someone who is not a resident of Qatar, but in the past I have successfully obtained India visas in Abu Dhabi and Riyadh without resident status.

I would advise you to find an agent (they sometimes call them mandoub) in Abu Dhabi who is familiar with all of the requirements and processes of the various embassies. You might also not achieve much in a week, since the process takes at least overnight in most cases, and visa issuing hours are usually only part of the day and sometimes not even daily. So it's easy to burn two days on one visa.

You might be better off just Fedexing your passport to an agent in Washington, D.C. and having him obtain everything you're looking for.