Yes, you may apply for a change of status to an H-1B if the job and employer meet requirements.

Your employer must file a petition during the H-1B cap window unless you're cap-exempt

You must maintain valid F-1 status until your new visa is approved.

Yes you can, although this is dependent on certain factors. However, it is important to under that if your visa is denied, you may have to leave the U.S. and re-enter on a new visa

There is the option of premium processing which can speed up the process if you have a limited time frame you're working with. However, this does not in any way guarantee an approval