It’s possible that some institutions are dictating blanket use of software, but where I am studying that’s not the case.

It’s basically up to the instructor on how to run the class.

Some require proctorio, some are on Zoom with lots of TAs to watch the screen, some issue an exam on your own time where the browser monitors what you do / visit, and others just offer take home projects.

No but it's very reasonable to wait a year before applying so that you don't pay out the nose for zoom classes (there are things like socialization and connections that make a super expensive college worth it)