Hi,
That control , controls all of the basic operations of your furnace. It sounds like it has failed, you have what is called a smart valve by Honeywell which are famous for failing also. The problem can actually be either part , it does sound like the black box in your case though. These were bad integrated controls and are no longer in use on new products. Everything you did makes sense , You can find them online if you want to change it out yourself. It sounds like you can handle doing the repair. Keep in mind there is a chance that you will end up changing the gas control valve also, if not now you will in the future.
http://www.famousparts.com/st9141a1002.html
Thanks
Mark
Journeyman Technician
UA Journeyman Pipefitter , HVAC, Refrigeration, DDC controls. 26 years.Commercial & residential