Well, that's a scattered mess!

For the dual monitors in the 2 rooms. If it is acceptable to have them turn on/off with the TV, then you can use a smart strip to control them with the power of the TV = [Link: amazon.com]

If you need individual control, then you can use IR controlled relays. Then have the relay switch power to the monitors (this will require splicing AC cords) = [Link: amazon.com]

Some multi-zone amps will have a wired IR input on the chassis, with discrete codes for each zone. Others might have a wired IR input on the back of the room keypads. You didn't give the make & model of this amp, so I'm going to guess we are dealing with a really cheap POS. So it has no special connections or controls beyond the keypads. That will make things difficult to control this.

Same with the HDMI matrix. Have no idea what we are dealing with and how it is controlled. So nothing I can suggest yet for that.

Are all the TVs the same brand? - so same IR code with work them all? or different brands and different IR codes? So 1 IR port could be used to control them all? HAVE to know these things.