thermostat not working?

[matter]

I have a new house, second winter, with a new Rheem furnace and new thermostat. One night furnace would not kick on. Flicked switch on furnace off/on, furnace kicked on and ran a few cycles then stopped again.

I changed batteries in thermostat, flicked switch on furnace off/on it ran fine for a few days then stopped working. I changed batteries again (just to be sure), flicked switch off/on, furnace worked.

Question 1) What is the probable cause here, a bad thermostat?
Question 2) Flicking the switch (on the side of the furnace) off/on always starts the furnace immediately. Is this a sign of a bad thermostat?

matter

[matter]

Additional information:
1) The furnace usually has a low hum, like a transformer. But when it stops working the humming stops. Flicking the switch starts the humming and starts the furnace. It's like the furnace shuts itself off.
2) This problem happended again last night - the third time in two weeks. It's always at night and always when the outside temperature is under 5 degrees. I live in Rochester, NY. I wouldn't think it would be possible, but could the exhaust be condensing in the chimney, freezing, and constricting the outflow of air? The chimney is regular stainless steel with about 5 feet of pipe extending above the top of the house. There is a stainless steel cap on top.

matter