Can a soldier be denied terminal leave for 30 days?

Yes they can. Not likely unless there is a unit mission that requires there presence.

Yes it can. Terminal/Transitional Leave is just a term used for the leave days saved up prior to ETS. It's still Leave. And still subject to the approval/disapproval of the Commander. It can also be yanked by the Commander even after it's been approved. I discovered that first hand in Germany many years ago.

Yes. But when in doubt AR 600-8-10 is always a friend. Make sure you out process completely as well. I personally was put in an AWOL status because I missed one signature. Leave was turned in, and I signed out, but the mistake wasn’t caught until about 2 weeks into my “terminal “ leave. I had to come back, serve the remaining days I was “awol” for. Then properly released. Just cross your t’s and dot your I’s. I had to wing it trying to final out. Hopefully you get more guidance. Good luck!

Are you flagged for anything or under investigation?

Leave is the COMMANDERS program. Yes it can be denied. If you have cleared installation and are otherwise waiting for your separation date, the commander can have you sit there, not on leave, until you separate