Red Vengeance, yes you (Martell) must kneel two influence to satisfy the cost of the event and your opponent will then satisfy the claim of that challenge. Ultimately you are transferring the responsibility to your opponent.

Drumbeater, the purpose of this card is to gain a power token if you win by 4 or more STR. For example if you have a total power of 10 and the opponent decides to block the challenge with a total strength of 5, you win the challenge and your opponent must satisfy the claim value. However, if you kneel drumbeater, it will be unopposed and you gain a power automatically (unless cancelled by an effect) and your opponent must still satisfy the claim value depending on what the challenge was (power=give up a power, intrigue=discard x amount of cards, military=kill x amount of characters),

Great questions!

Have fun!

1 - Red Vengeance does two things; the first is cancel the regular effects of claim, the second is choosing an opponent to satisfy the claim instead. That opponent can be (and in joust, invariably is) the attacker who just won the challenge.

2 - A challenge is normally considered unopposed if there are no defenders, or the opponent counts 0 str in the challenge (say, if he blocks with characters you get to ignore the strength of because of intimidate). What Drumbeater does is allow you to kneel him to make it so that if you win the challenge by 4 or more (Say, your 7 str fleet to his 3 str defender), it'll also be considered unopposed regardless of defending strength. This is particularly interesting out of Greyjoy who, in addition to getting the unopposed power, tend to have a lot of other effects that key off winning unopposed challenges (eg. Victarion's Scheme). The timing for this decision is a "challenges" action, so it can be taken either after declaring attackers but before stealth targets are chosen, or after defenders are declared but before resolution of the challenge starts.