Inheriting stocks and selling them

My father had a stock account using his SSN, showing both his name and mine (right of survivorship). He just died. Am I allowed to sell the stocks now, as the account stands vested? And - are the proceeds from the sale taxable on my taxes? On his when I do his next year? Do I figure capital gains on the difference between the stocks' value on the day he died and the day I sell, if the proceeds are taxable?

 

There is no inheritance tax in his state nor in mine. Thank you.