So assume he says "Either before you entered the room I ran a simulation of this problem as presented to an agent running TDT, or you are such a simulation yourself and I'm going to present this problem to the real you afterwards", or something similar.

...

Well, a TDT agent has indexical uncertainty about whether or not they're in the simulation, whereas a CDT or EDT agent doesn't.

Say, you have CDT agent in the world, affecting the world via set of robotic hands, robotic voice, and so on. If you wire up two robot bodies to 1 computer (in parallel so that all movements are done by both bodies), that is just somewhat peculiar robotic manipulator. Handling this doesn't require any changes to CDT.

Likewise when you have two robot bodies controlled by identical mathematical equation, provided that your world model in the CDT utility calculation accounts for all the known manipulators which are controlled by the chosen action, you get correct result.

Likewise, you can have CDT control a multitude of robots, either from one computer, or from multiple computers that independently determine optimal, identical actions (but each computer only act on a robot body assigned to that computer)

The CDT is formally defined using mathematics; the mathematics is already 'timeless', and the fact that the chosen action affects the contents of the boxes is a part of world model not decision theory (and so is the physical time and physical causality a part of world model not the decision theory. Even though the decision theory is called causal, that's some other 'causal').