First, wow! This has to be an incredibly painful situation for you! Hugs for that.
Now to the hard stuff.
You seem to have two problems: Mark and Cathy.
From what you're writing, it's pretty evident that Cathy is what some poly circles call a "cowgirl"- someone who pretends to be poly or open to poly but with the intent of actually splitting you from your partner so that she can have him to herself. That's bullshit.
Right now, you should not be negotiating with Cathy about your boundaries with Mark. She's not Mark, and he's the person you should be talking to about that. Plus it's clear that she has no respect for your relationship with Mark and is a pretty poisonous presence. Quite frankly: fuck her. And I don't mean that in the fun way.
But I think she's the least of your problems here.
What you've described in this post brings up several red flags for me:
* Mark wants the freedom of outside relationships for him but isn't allowing you that same freedom- essentially asking and practically forcing you into making a sacrifice that he's not willing to make for you.
* It seems like this Mark/Cathy thing was already going on to an extent behind your back. If he's jumping into a date two hours after the discussion, it's clear the date was planned before the discussion. That's definitely the WRONG order and it seems to suggest that your feelings are not high on Mark's priority list.
* Mark now seems putting you on some kind of romantic probation, basically saying he's not sure if you're good enough to stay with. That's a bullshit thing to do. And he's leveraging his relationship with Cathy against yours.
All of these things suggest that Mark is leaving little room for your needs or feelings in this relationship. For me, these would be pretty fundamental issues and I really can't see anything else that a person might offer me that could outweigh these REALLY big problems. So I have to wonder why you're choosing to stay in a situation where you're clearly being treated pretty horribly.
If my partner was acting in ways that clearly showed that my needs and feelings are not important to him, I would set a very clear boundary with that: I would leave.