Applying for Green card while on F1 OPT-STEM extension

If you qualify for employment sponsorship in the second preference category, the timeframe might work out. You would want to discuss the specifics with an immigration attorney as much would depend on your background and the company sponsoring you.

Yes, that is (very) possible. With proper planning and impeccable execution that plan might work out very well, and of course, applying in the EB-2 category, if both you and the job offered qualify will be a great enabler. Make sure to only work with a highly qualified attorney on this one. No less.

Your employer should be advised to engage a reputable immigration attorney and provide you with access to that person for questions and clarifications. In general, there is no connection between your status and employer's immigration petition per se.

It may be possible depending on your country of chargeability and the category in which the petition will be filed. It will be best to work with an immigration attorney to ensure all bases are covered.