Class 2 transformer wire (only) through electrical panel?
Two long as a comment, 
If you are in the US they are called legs as residential is normally a single phase feed that is center tapped or a split phase. 

Don’t expect them to be identical they will be close usually within 1% unloaded but not identical depending on loading I have seen them +15% off with no way to balance due to a single intermittent large load.

You can split a duplex receptacle to have each on its own circuit this requires handle ties at minimum and can use a common neutral with current code a double pole GFCI would be needed to feed these receptacles in most locations. 

You can run your control wire or low voltage through the panel if the wiring insulation is rated for the voltage (your example of THHN is 600v would comply with code)  but the reason we usually do not is because of the electrical noise generated with the 120v circuitry and the low voltage separation that you have identified.

I hope I have hit all of your points , to get better answers a single question would help.