Sorry doc, been there, got the T-shirt. Have been to all Gulf States and many other countries since I first went to the Gulf in 1978.
As one of the Canadians indicated on another thread. You are at half-way house in Qatar so one of the benefits is being able to visit other countries.
Sadly, companies now only provide excursion tickets for annual leave when previously they provided full economy which made it possible to travel almost free.
Yes exit permit or visa from sponsor