In the question about cards, the binary observation is absolutely correct but gives the impression that you need more "structure" to solve the problem than you really do. I prefer (even though it maybe yields a textually longer proof) to do it this way: sort the possible configurations lexicographically ("dictionary order", where things further left always take precedence) with up < down; and then note that every operation you do moves your configuration earlier in the order, at which point you're done.

... Oh, and the problem as stated above isn't quite right. In the configuration UUUUUUUUUUUUUUUUUUUD there is no legal move according to your statement of the problem, so you don't terminate with all cards face up. More generally, your procedure always flips two cards at once so the parity of the number of D cards can't change, so half of all starting configurations are unable to end with all cards up and in fact will end with UUUUUUUUUUUUUUUUUUUD. (To fix this, either just ask for a proof that the procedure always terminates, or else make flipping the card to the right of the one you turn face-up optional.)