Sounds like you have a poor 12V connection from the fuse box to the furnace.
When you ran the jumper wire to test the thermostat and A/C, did you also connect the 12V wire from the furnace up there at the thermostat to the jumper? If there is a poor supply wire to the furnace that would send power from the jumper back down to the furnace to override the original connection. If that works you need to find the bad connection from the fuse box to the furnace.
Did you try running a jumper wire from the fusebox to the furnace instead of using the one that pulls down to 1.5V?
For the wire nuts you tightened, did you remove them and inspect the wires? They should never need tightened and I suspect the wires may be damaged. Make sure all of the strands are intact, particularly where the enter the insulation. If you pull the wire nuts you may need to cut the ends and strip them back a bit further to ensure a good connection. The wire nuts do not make the connection. Twisting the wires together is supposed to do that and the wire nuts only hold the twist in place.