I seriously would suggest a professional installer to help you they would have the experience to calculate throw distances to choose the right projector, and mounting distance, and the right cable options.
However if you really want to diy it here are some things to consider.
Mount companies offer options that allow you to use any standard metal pip to custom cut the height off the roof. The mount attaches to the pipe by being threaded around, like a coupling. This is not subjected for an active gym where flying balls could knock the projector out of alignment.
Another option would be to mount the projector on an opposite wall, which is what I suggest for gyms. By using the right
lens choice for your projector you can achieve some super long throw distances. You might also consider directly mounting the projector to the ceiling and key stoning the image to reduce cost, and keep the projector height off the ground to avoid balls.
as far as cabling, I would sudgest using the bulans if you’re going over 150', however composite, component and vga I have been able to achieve good quality picture on high quality cables over distances of 300' however that’s pushing it.
I would suggest cat5 for control, or an ir blaster.