The wyze gets it’s own power from Rc and C, so technically, if you connected one of the normally switched lines to C, there might be enough juice to power the wyze and the other load, although putting them in series like that might drop the voltage to the point where it doesn’t work quite right. Normally the loads are in parallel, to minimize voltage drop.
For your second question, the wyze has internal circuitry that detects if something is connected to Rh and switches the heating loads to it if there is. The cooling loads stay on the transformer with Rc, which is not connected to the other (common) side of those loads, and thus a full circuit cannot be made.
Furthermore, some people have attempted to connect the C from the transformer to the wyze and have then spliced the other wire into Rc, which creates a hazardous situation. Just because 2 transformers have a 24 volt difference between the leads does not mean that they are both the same potential to ground. Often, the transformer in your furnace will have one side directly connected to ground, however the wall adapters that people are using will usually be floating, and may be several hundred volts to ground. They are both still 24V between the leads, but in reference to ground they are very different and cannot be connected together.
If you plan on using a system like this, a normally open isolation relay is required on the furnace end. I’ve added a diagram to this post to show what that would look like in a 2 wire run, with the wyze getting power from a wall adapter and switching an isolation relay. This is commonly used if you have an aquastat with T T connections, as they usually do not provide enough current to drive a smart thermostat, however it could also be used to run cooling on a system when you are using an external transformer, if you had enough relays, because you would be able to use the R wire that normally runs between the furnace and the thermostat as the C wire for all of the relays. The limit then becomes the VA rating on your external transformer.
Heating with 2 wire run and non-24V equipment:
In addition, if you have 3 wires in your run, and wanted to not have a wall adapter running to your thermostat with an aquastat, you could put the external transformer in the same room as the aquastat like this:
