If I kill Al-Sedexus (I understand you can; haven't gotten around to it yet and I really just got into the castle and got the custom items from Ferro) would it somehow "clear" the Rapax? Well, what I mean is that since I still have not gone to the Rapax Away Camp nor explored it, would that place disappear or just not turn out right if I clear the Rapax Castle before I clear the Away Camp?

Nope - nothing "clears" the castle that I know of. Becoming a templar makes them friendly (until you clear the away camp). Killing A-S after becoming a templar doesn't make them hostile. Freeing prisoners at the away camp - that pisses them off.