I'm not there remember to see what and/or how you are testing or anything.
The results of the tests don't make sense, so lets go another route and try to eliminate this.
Let me explain. You have BOTH modes on at same time. You disconnect White wire at Furnace and the heat stops, reconnect and the heat starts again.
Yet, when you tested the White wire with a good ground, you did NOT get a reading.
By what you stated, the heat was receiving its power from the white wire, yet there was no voltage reading when testing white wire.
Set t-stat to Cool and Auto and 65*.
I assume BOTH Cool and Heat comes on here.
By switching t-stat from COOL to OFF, do BOTH cool and Heat shut offf ?
You must also realize that this may not have anything to do with the power interruption.
It sounds more like two wires becoming bare and touching somewhere in the circuit, most often caused by rubbing against the sharp edge of something or being chewed on my mice.
This will cause a back-feed sometimes also and could explain the non reading when you disconnected the white wire from furnace.
Also, the Red coming from the transformer to the t-stat does NOT
come from the common side of transformer.
Where/what was the White wire connected to at furnace where you disconnected it ?
Some people are Humbly Grateful, while some are Grumbly Hateful.................... Which one are you ?