I believe the intention is to take RATED CAPTAINS only. There will be no F/Os and this is supported by the statement on the recruiting agencies website.
The Amiri Flight has nothing to do with Qatar Airways, other than on a 3rd party provider basis. They are two totally separate entities so no transfer between either parties is possible.
The C17 is not part of the Amiri Flight operation.
Transfer between fleets within the Amiri Flight generally does not occur, but I believe the policy on this is not fixed. Given that the salary is identical, regardless of what aircraft you fly, there is no financial incentive to move fleets, nor would there be much company incentive to do so as it would be an unnecessary cost burden.
The CL300 and CL605 belong to Qatar Executive which is a commercial operation and is part of Qatar Airways and has nothing to do with the Amiri Flight.
The Amiri Flight is an entirely private operation. It is advertised as an "Airline" on the recruiting website, but it is purely a private operation.