Khal Drogo

How does his effect work with the event "By Sword or by Guile"? Since Drogos effect already cancels the normal claim, does the event cancel Drogos claim effect or do both effects remain active?

Actually, Drogo's effect does not cancel the claim effect, since the description does not use the word 'cancel.' Drogo's effect replaces the claim effect of the challenge, like with Pyat Pree (QoD). By Sword or by Guile does cancel the claim effect and then instigates a card effect of moving 3 power, so the way these would interact is: win challenge with Drogo alone, claim effect changes from whatever standard effect is to his specific claim effect. You may then choose to play BSobG and cancel his claim effect to move 3 power. If you chose to activate BSobG first, then the normal claim effect has already been canceled and Drogo's effect cannot be activated.

Note also that since Drogo's effect only replaces 'the normal claim effects,' cards like Battle for the Shield Islands which have already replaced that normal claim effect prevent this ability from being used, while BSobG can still be played here since it is only replacing the claim effect.

Khal Drogo doesn't cancel the claim effect. (The word "cancel" doesn't appear anywhere in his text.) Rather, Drogo replaces the normal claim of whatever challenge he attacks alone in so that it resolves the way his card says. If you use By Sword or By Guile to cancel the claim effect, it never resolves in the first place, so it cannot resolve in the "new" way indicated by Drogo's text.