Can I sell unvested stock?

Typically no. If it is RSUs given by company, you do not own any shares until they vest. Only after the units are vested, you can then sell the shares. And they vest over a period of time.

If you leave the company before the vest date, all unvested units are forfeited. Incentive from the company to keep you around. Or you can negotiate with the new company to cover some of the loss as part of your new package.

You technically can but it is called naked short.

If you naked short, you have to cover or buyback by the end of the day or you will face a penalty and fine.

If not you can use CFDs to properly short sell a stock that you do not own.