No Prince William is not The Prince of Wales. That title is held by the eldest son of the reigning monarch. The reigning monarch is Queen Elizabeth II and her eldest son is Prince Charles, who is...
Yes. He is also heir apparent of the heir apparent, so he is expected to become King one day, assuming the monarchy doesn't end before he gets to reign.
Their surname, by law, is Wales. The last name of the two Princes has always been under some discretion. This is because their grandmother, Queen Elizabeth, married Prince Philip, Duke of Edinburgh....