All FrontRunner stations have overnight parking except Salt Lake Central and Layton (daytime only) and North Temple (none). I used to reverse commute from downtown to Orem and left my car at Orem Central on weeknights.
Posibaly
I've personally left my car multiple times. As long as you don't leave anything of value in view, you should be fine.
I've done it several times for multiple days without any problems. There is actually a UTA Police Station there and you might feel a little safer parking next to their building.
Depends on the car. I would not recommend it. If overnight maybe the back of the hospital parking lot that has more activity would be safer.
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