"Салата з сиром прошу." is also a correct way of asking for a salad with cheese, please. ;)
Салата is wrong. Салат is inanimate, so in Accusative it does not change the form. If it was something feminine, e.g. канапка з сиром, you could say канапку з сиром as well, as a request (e.g. to the waiter)