Don't expect that it's going to be feasible to fly the aircraft to a maintenance base that's more than a handful of hours away. Typical GA aircraft require at least a 50-hour check, which means it needs to go to a maintenance facility every 50 flying hours. If you spend 20 of those flying hours just flying back and forth to the maintenance facility, it's going to be very costly. And that's not counting fuel or the value of your own time.
I could not find a great circle calculator quickly enough but by road Qatar-Paris is 6000+ kilometers. Say 3500 nm by air, which means 35 hours flight time at 100 knots - a fairly typical cruise speed for a GA aircraft. Fly faster and you can maybe reduce it to 25 hours. Which means just Qatar-Paris and back for maintenance will require 50 flight hours at least. It will be more cost-effective to find an engineer in Paris and fly him to Qatar, all expenses paid, to perform the maintenance on your aircraft.
As far as aircraft and pilot licenses is concerned, the general idea is that you need a license issued by the country that registered the aircraft, and then you can fly worldwide. So for an N-reg (US registered) aircraft you need an FAA license, for a Qatar-registered aircraft you need a Qatar-issued license and so forth. You can essentially pick whatever combination works best for you, and I agree that an N-reg plus FAA license would be the most sensible in your case.
Can you base a non-Qatar registered aircraft in Qatar? I don't know, but generally I know that if you base an aircraft in a certain country, there are requirements and limitations. For instance, even though you can base an N-reg in Europe without having to re-register it, you still need to pay VAT (or be exempt), and EASA is also continuing to make noises about requiring pilots to hold an EU-license in addition to their FAA one.