Donk is a (potentially offensive) term used to describe a very bad poker player. A donk bet is a bet made out of position after check/calling on the previous street. The link here is that it's usually 'donks' who donk bet hence the name behind the bet.
An up-and-down bet involves two selections and two separate bets on both those selections, in each of which the second bet is conditional on the first one winning.