My doctor is too busy and 'forgets' to put requisition through? Can I just schedule my own test in radiology?

Call. His nurse to help schedule test. If urgent go to er.
Depends. It depends on the state and exam. In ny a prescription is needed for a exam.
No. You need requisition to permit facility to perform radiology study.
No, but... A patient can call a radiology department and tell them that their physician would like them to have a test, but that he has not put in an order/requisition. The radiology department will often track down the orders for you.