If you have tmj, does keeping your mouth open for a dental cleaning make your TMJ even worse?

Possibly. You don't really want to avoid the cleanings and check-up if possible. Sometimes the hygienist can provide breaks and cautiously avoid you opening your mouth too wide. The best advice is to try and get the proper tmj/tmd treatment. Sometimes some physical therapy can be utilized (ultrasound therapy, tens) prior to and after the appointment to help you. You might want to consider sedation.
Maybe! Yes, TMJ problems may be temporarily made worse by having to keep your mouth open wide for a dental cleaning. But, that doesn't mean you shouldn't get your teeth cleaned! just tell your hygienist about your TMJ problems and he/she will take special care to avoid straining your jaw or jaw joint. If your TMJ problems do flare up, it is likely to be temporary and should respond well to simple things.
Yes. Prolonged periods of mouth opening, especially excessive opening can aggrevate a pre-existing TMJ problem. It is reasonable to ask your dentist/hygienist to shorten appoinment times and minimize protracted periods of mouth opening during cleaning and other dental visits.
Yes, temporarily. Yes, it can temporarily. Try to have your teeth cleaned in shorter periods of time. Maybe break it down into 2 sessions.
It can do. Please do not let that fact tat you have a TMJ problem prevent you from getting your teeth cleaned properly. Have your dentist address the issue head on and get the relief that you need for the dysfunction so that you can receive the care you need. It is more likely that the process of holding the mouth open for cleaning only exacerbates the problem, rather that making it worse.