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.

Yes, you may study on H-4 visa. You need not apply for F-1 student visa.

No, you may not work on H-4 visa in the U.S. The principal H visa holder has to support you.

Your H-4 status depends on the status of the principal H visa status. You lose your status once the H loses status.

On H-4 visa, you may:
   
  •     Enter into the U.S. along with your spouse or join him/her later
 
•     Not have to maintain a foreign residence
 
•     Travel in and out of the U.S. or remain in the U.S. continuously as long as you are in the status
 
•     Study in any U.S. university
 

This visa is issued by the U.S. Citizenship and Immigration Services (USCIS) to immediate family members (spouse and children under 21 years of age) of the H-1B visa holders.

USCIS allows immediate family members of H visa holders (H-1A, H-1B, H-2A, H-2B, or H-3) to get H-4 visas to lawfully come and stay in the US. These visas are usually issued at the local US consulate office abroad. However, if you are already in US, you can obtain H-4 visa by filing Form I-539 for change of status.

H-4 visa holders are not allowed employment in US and are not eligible to get a Social Security Number and cannot be employed, but can hold a driver’s license, open bank account(s) and get ITIN for IRS Tax purpose.

Family members may alternatively be admitted in other nonimmigrant categories for which they qualify, such as the F-1 category for children or spouses who will be students or the H-1B category for a spouse whose employer has also obtained approval of an H-1B visa petition to employ the spouse. An H-4 visa holder is admitted to the U.S. for the duration of the primary (H-1B, H-2A, H-2B, or H-3) visa validity.

Since H-4 visa holders are not issued an Social security number, an ITIN (Individual tax identification number) should be obtained before filing for join tax returns. There is a form called w-7 that needs to be filled and related documentations which is mentioned in the w-7 form must be notarized and to be sent to the address mentioned in the w-7 form.

The notarization can be done in any of the UPS branches in the USA, for this you need to take the original documents that are applicable for you which is mentioned in the w-7. The notarized copy should be sent along with the filled w-7 form to obtain an ITIN number.

This number needs to be obtained before sending the tax return for the joint filers of income tax. Otherwise wife’s exception will be rejected and refund will be less.

Make sure you receive this number before filing for either federal or state taxes for joint income tax return filing only.