You can’t choose to discard a card at random if you have no cards in hand. If you have one card in hand, you may discard it at random, even though that’s not very random.

Once three or more creatures you control have attacked, you’ll be given the option to discard a card at random, even if some or all of those attackers leave the battlefield before Gallia’s triggered ability resolves.

While resolving Gallia’s last ability, you can’t discard multiple cards to draw more than two cards.

If Gallia leaves the battlefield, Satyrs you control lose haste. If they haven’t been under your control since the turn began and don’t otherwise have haste, they can’t attack this turn. If they’ve already attacked, they remain attacking creatures.