Can Eldrazi Processor effects "counter" madness?
Yes, you can remove a card discarded through Madness from exile before your opponent has a chance to cast it.

As per the the relevant rule you already quoted, Madness has two parts to it.


  702.34a Madness is a keyword that represents two abilities. The first is a static ability that functions while the card with madness is in a player’s hand. The second is a triggered ability that functions when the first ability is applied. “Madness [cost]” means “If a player would discard this card, that player discards it, but exiles it instead of putting it into his or her graveyard” and “When this card is exiled this way, its owner may cast it by paying [cost] rather than paying its mana cost. If that player doesn’t, he or she puts this card into his or her graveyard.”


The first part says that the card with Madness is put into exile rather than the graveyard if it was discarded.

The second ability triggers off the card getting discarded to exile by Madness. Only if and when that second ability resolves can that player cast that spell for its alternative Madness cost. You can respond to the second ability by removing that card from exile with your Displacer/Processor combo. Since that causes the card to change zones and become a new object, the triggered Madness ability will lose track of that object and your opponent cannot cast that spell when the ability resolves.