"It would not turn on with the auto setting, and cool air came out when set to on."

FYI: The Auto/On switch only controls the fan, it does not control the heat. When set to Auto, the fan turns on when the thermostat calls for heat.

"What could be causing the heat to not work overnight like that and suddenly come back on?"

Nowhere did you write that the heat ever worked.

Like already mentioned, the auto/on is for the fan only. if its in the on position the fan will run all the time. If its in auto, it will run when the unit calls for heat or cooling. Make sure the thermostat is set for heat and set at a temperature higher than the room temperature.

If your thermostat shows return on it, it may be programmed for some type of eco schedule. i woukd try to get into the settings for now and turning that function off until you can verify your furnace is operating properly.

Some thermostats require batteries, which of course need occasional replacement. Have you checked on that?