Heat is not responding to thermostat set temp but comes on when turned off and back on with the thermostat switch

In the heat mode the larger copper pipe going to the AC unit outdoors should get very hot, if not then the refrigerant might be low or the reversing valve which switches modes from Cool to Heat may be bypassing. the fact it shuts down tells me its getting overheated likely due to low refrigerant level.

Warm air rises, not vise-versa. If you are running the A/C on the first floor and leave the upstairs A/C off, then the downstairs unit should not be working all that much harder than it would if the upstairs unit was cooling the upstairs. So no, cold air is not drawn upstairs by thermal convection. That does not really happen.

Upstairs unit off and downstairs unit at 78 should reduce your energy bill versus running both the upstairs and downstairs units.

Edit:

There is an exception to what I said above. Because the upstairs unit must handle the extra heat load that comes from the sun shining on the roof, it is often the case that for equal square footage, the upstairs unit is larger capacity than the downstairs unit. Also, the cold air from the upstairs unit will flow down to the first floor via thermal convection (cold air sinks). If your downstairs unit is a bit undersized and your upstairs unit was meant to pick up the slack, then your downstairs unit will not be able to keep the downstairs cool all by itself, but this fact will not make your utility bill any higher than if the upstairs unit was running (unless your downstairs unit is also using large amounts of power from a malfunction and needs to be repaired).