QR will give you four months notice before your joining date. All license and government medical screenings are done after you arrive. Priority is given to the government medical since this is the first step in getting your Residency Permit, RP. Without the RP you cannot apply for a driving license or internet, etc. The RP process can take a month. The aviation medical is more of a logistical exercise and not very tough. If you have no issues on your home medical you will not have any issues here.
Bring an International Driving License! You can rent a car and drive it on an International Driving License for up to six months after joining but only for one week on your national license.
Organize a rental car before joining. I can recommend Strong Auto but there are many others. Renting a car is very easy and will cost you around QAR2,000/month. Do an internet search. You will only get company transportation for the first week, after that you must have your own car. Many guys share a car for the induction and ground school phase when everyone has the same roster.