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Carriers will run fan after multiple retries. Check limit switches and pressure switch.

Never messed with Noma t-stats but have with Rite Temp. They have very poor customer service and you have to read instructions well to know what jumps to move where. Like NorthernHeat said Honeywell...

Not usually but with any thing man made they can. I'm a tech and have seen them fail in days.

Read all post but not understanding which vent you replace water heater or furnace. Water heater vent looks like a dual vent ,intake and exhaust, and would be fine if outside is not completely sealed...