We just use accusative when asking for things. You can consider that this sentence has the verb «дайте» 'give' dropped, if that helps.
No. Cup is ча́шка, while glass is either стака́н (bigger glass, without a stem), бока́л (stem glass) оr рю́мка (smaller wine glass).
(Ча́шку is the accusative case of ча́шка. Accusative case of рю́мка is рю́мку. Бока́л and стака́н don’t change their form in the accusative case, they remain бока́л and стака́н.)