EDIT: Defeated. Clearly I need to bookmark the Comp Rules. :-P
How does madness and graveyard removal work can I deny them their madness spell with it?
Madness does not put a card into the graveyard, so you cannot prevent them from casting it by removing it from the graveyard.
More on madness can I stifle a madness card?
Yes. But if I'm reading it right, you have to Stifle before they pay the cost. Someone else could back me up or correct me?
Academy rector can I remove her from the graveyard before my opponent can use her ability?
Yes. The triggered ability goes on the stack and you can respond to it, making it impossible to fetch an enchantment with it.
More on rector can I stifle her ability?
Yes.
How does stifle and illusionary mask interact, can I do anything to interfere with the mask player or the face down creatures?
You can Stifle the ability, but the creature remains in their hand. Turning the creature over can happen at any time for no cost. I suppose you could Stifle it, but they can do it again without a hitch.
As well do you draw a card from every card that has cycling or only the ones that actually say draw a card?
Cycling always draws you a card. If you mean that some cards don't have the reminder text, italicized text is irrelevant for rules purposes.