no they would not, but they would certainly investigate you and look deep into your life and relationships with others.
If your company respects law approach, all 3 answers is "no". But anyway, unpleasant outcome for you is very possible.
Lots of mis-information here based on a lack of information provided by the OP. A lot of companies restrict personal trading. This is for several reasons: 1. they don't want you to trade on client/customer information (typically law firms, consulting firms, audit firm, financial service firms, journalists, government employees) 2. they don't want you to compete with their business (typically financial service firms, mutual funds, hedge funds, etc.) 3. they don't want you to be distracted from your core job which could be related to investments (typically financial service firms, journalists, government employees) A lot of companies will restrict trading in their own stock around earnings periods because they want to limit the risk of insider trading. You may not have inside information but someone at your level in another department might so they ban it for everyone during a blackout period. These rules will be highlighted clearly in your employee handbook. If it's not there then it's fine. Employee personal trading is a needless risk for a public company and their lawyers will ensure it's handled properly. If it is a gray area, someone in HR will be happy to find the answer for you. If you violate these policies you run the risk of termination. For what it's worth a lot of employees hedge their restricted stock. The most famous was Mark Cuban who collared his Yahoo stock after selling Broadcast.com. In my old job we had clients who sold businesses to public companies, and who owned sizeable (billion dollar positions) in those public companies who would hedge out those stock holdings with collars and other structures. If you are going to hedge restricted stock, remember that it's not your stock yet and you may only receive it under certain conditions. So you might end up having a speculative position instead of a hedge.
Just ask HR for the company policies about trading in the company stock. If there's no rule against shorting, go for it. If there is, you've got to ask yourself if you want to risk your job and whether you think they would have any way to find out if you had shorted it. Certainly in a regulated industry like finance, if they're getting your brokerage statements you better not use that account.
Some companies have a specific policy about such and others do not. I've seen companies have policies about "when" such can occur. For example, you already said you "cannot sell it until the vest date"...that implies they have a specific policy about such and that you're aware of the details about that policy. Also, I know someone in a current situation like you. Instead of shorting his own company stock, he shorted a competitor of the company that was just as over-valued. Anyways, both companies have been soaring lately...he's lost a bundle.
Why don't you just buy puts? I've done that before. I've got RSUs that vest every July. I buy put options to cover the risk in Feb or so of that year.
Re #3. Yes you can be fired for that. Unless you have certain union protections you can be fired because your boss doesn't like you or because you wore a red shirt to work or for no reason at all just as you can quit for any reason you'd like. There are certain exceptions of course. You can't be fired for the big five - age, race, sex, religion, national origin and there are a few other exceptions. But, in general, yes to #3.