QIf my opponent's hand is empty, can I still play this spell?

AYes, you can.

QHow do I discard a random card?

AFor example, if you have 6 cards, you could assign each card a number from 1 to 6 and then roll a 6-sided die. If you have 3, you could assign two numbers to each card and then roll. If you have two, you could assign three numbers or flip a coin. Alternatively, you could shuffle your hand and simply have someone else pick a card.