As soon as your RP gets cancelled, there is a specific period of time within which you will need to exit Qatar. Previously, it used to be 7 days but lately I heard you can stay for a couple of months from the date of cancellation of RP after which a penalty will be levied for each extra day of stay. You can check on this required grace period and make sure for yourself. Your company's PRO should be able to tell you about this.
As for your queries for the other issue, You can keep you bank account active even after you exit Qatar or close it on the last working day of the bank before your departure. You can keep your drivers licence with you till it expires. Your apartment rent is between you and your landlord and has no connection with your RP or your exit. It is advisable you sort out this matter before you exit. It is also advisable you sell off your car before you leave on your final exit.