What the law says and enforcement of the law are sometimes 2 different things, but generally employees should not be shorting their own stock, lock up or no lock up.

The issue is that this creates a conflict of interest.

In the worst case scenario, as an employee, you can literally do a bad thing to cause the stock to go down. E.g. An engineer can make a bug which blows up a rocket.

So then you could short the stock, bug a rocket, and become super rich.

It's the same issue with athletes betting on their own team - it's trivial to throw the game.

My holdings aren't even locked up, and I'm still not allowed to short my employer -- true as a matter of policy which could get me fired, and true from a US legal perspective most of the time given my role.

Almost certainly, outside of standard trade restriction windows. I don't think they have any control over what you do in the market outside of preventing insider trading.

No. Typically lockup agreements prevent any kind of trading of derivative or synthetic positions (think: shorts, swaps, options, etc).

No. Especially not if you have friends who could short the stock and you come to some sort of pocket agreement that never sees the light of day. Especially then.