What you say is a bit ambiguous. You say that students may have to record themselves during the test then you say you'll stop the mic and webcam.
The test may happen as a conference so the teacher can keep an eye on you real time. If so you'll need the webcam and mic and also ToR may get out of question for making the things too slow.
I know in my country, universities are switching to oral exams over the webcam, open book exams, and physical exams with respect to social distancing rules.
Luckily, I already know how the exams are going to be. Some will be multiple choice tests, and others are going to be developing exams that later on have to be scanned and sent. In both cases, there is no real need for such a strict control, and if someone is going to cheat, there's really nothing stopping them from doing so. Getting not very creative, any pdf file can be searched so it'd be pretty easy to use them. Apart from that, it's well known that these exams are not meant to evaluate our knowledge, but to evaluate our memory....
About distro, since you'll use a VM and will not "touch" your main data (that's smth you set from VM, not the OS inside VM), you can consider that VM disposable and it doesn't matter how much privacy-oriented will it be, it doesn't matter.
Make sure the VM OS will have installed everything you need!!
So the data from the computer should be safe.
The content of your room may be somewhat visible, I recommend you change the position of your office so your back will be next to a wall.
The IP however I don't know how you can hide without losing speed.
The whole point of using the VM is to not give it the webcam hardware; in case they make me install some extension that will search for it. You can't use something that you don't have. I'm not really concerned about loosing browsing speed, and it could be benefitial if the get too stupid with the whole streaming thing. A public university should not assume that people have high-end internet data available at home (and many other things), but that's another whole topic.
This one is pretty tricky... Do you want to risk not getting credits because you don't want to give a virtual tour, or because you decide to run inside a sandbox, or use tor?
I realize that, as a human, you have the right to privacy... But i'm not confident a university will give you credits if they have the slightest idear you might have cheated. Eventough i graduated ages ago, i still think i'd put my grades before my privacy and "sell out", comply with their demands and just get it over with.
it's not really about risking getting evaluated, the things I've seen here during the years I've been in this university would amaze many people. I've seen people's mark rise from a 2.3 to a 6 after scanning a 600 page book to pdf; and that's one of the least atrocities I've witnessed.
I understand that the university wants to ensure what the call "quality", but it's really annoyin that they are getting all worked up and worried about it now, when they don't give a shit during normal conditions. Yes, I agree that some sort of control is required, but the range of that control is just too wide. Another university that is close to mine, is making students do the room-tour, and then place the phone in a strategic place that will let stream the whole "working area", while at the same time, they have to get a webcam on with a screen recorder, it's pretty crazy.
My goal is to comply with the rules that are set for the evaluation, but not giving them more than the bare minimum requiriments. There are some lines that cannot be crossed, and even if some teachers tend to forget it, students have some rights too. Privacy is one of them, and I'll fight to protect it.