No, you may not apply for extension of stay on D-2 visa. The USCIS does not authorize extension or change of status for D-2 visa holders.
No, you may not study on D-2 visa. However, you may engage in employment in the U.S. only for the vessel for which you have got the visa.
Note: The USCIS has redefined the term 'employment' with particular reference to the employment of crewmen. The term 'employment' now means any service or labor performed by an employee for an employer within the U.S., including service or labor performed on a vessel or aircraft that has arrived in the U.S. and has been inspected, or otherwise included within the provisions of the Anti-Re-Flagging Act, but not including duties performed by nonimmigrant crewmen as defined in the INA.
As the D-2 visa is a single entry visa, you must reapply for the visa each time you enter the U.S. because according to the immigration laws the D-2 visa expires the moment you leave the U.S.