Heater issue

A furnace will run blowing cold air if the pilot light is not lit. The issue regarding reaching temperature and running then shutting off could possibly be a faulty thermocouple. They can be purchased at any home center for a minimal amount. Shouldn't be more than $10 or so. There is no way the Gas company is correct saying it is a dirty filter. Another issue that could cause the unit to shut down would be the exhaust fan which vents the fumes from the furnace to your stack (double walled vent pipe) having an intermittent problem. Cost is @ $150 or so depending on your location, age of furnace, time od day or night if an emergency call is needed. There is a safety shut off switch behind one of the panels on the unit. When you remove the cover to light the pilot light the switch is in plain view. It could either be sticking or faulty. It will prevent the furnace from running and will have nothing to do with the temperature, so if the furnace is running, it may have nothing to do with it. But if it runs and then shuts down, it is possibly a problem. I hope this helps and good luck with your problem.