Yes, you may apply for extension of stay if you need extra time to complete your studies. Your Form I-20 can be extended by the designated school official. The I-20 extension should be filed along with your passport, Form I-94, Arrival-Departure document, a letter stating the reason for your extension.
Yes, you may apply for change of status while on F-1 visa. The most preferred change is F-1 to H-1B. You may avail of other options like M-1, H-3.