Wow, a lot of people are following this. The C-17 is a cargo plane. While Qatar has a military, it's not huge and this plane isn't about bombing villages or nuking things from orbit. read more
The Qatar Amiri Flight is a VIP Airline owned and operated by Government of Qatar along with Qatar Emiri Air Force. The vast majority of its fleet of Qatar Amiri Flight and Qatar Emiri Air Force are painted in the standard livery of the commercial flag carrier of Qatar, Qatar Airways. read more
The vast majority of the Qatar Amiri Flight fleet is painted in the standard livery of Qatar Airways, like this A340 below. The aircraft is only operated by the Qatar Amiri Flight, not owned by it. The picture below shows the Qatar Air Force roundel on the underside of the wing. read more
Probably because the C-17s will be useful in logistical support for UN ops around the place (ie. great international PR); no other Arab state has a similar capability (local one up-manship); helps Boeing keep production open and therefore helps the US forces (good for staying in with the Yanks); and because it is always useful to buy your principal benefactor's expensive toys. read more