Your H-4 status depends on the status of the principal H visa status. You lose your status once the H loses status.
No, you may not work on H-4 visa in the U.S. The principal H visa holder has to support you.
Yes, you may study on H-4 visa. You need not apply for F-1 student visa.
Yes, you may apply for extension of stay on H-4 visa. H4 extension may be granted as long as the principal H visa holder maintains status.
Note: Your status depends on the status of principal H visa status. You lose your H-4 status once the principal H visa holder loses status.