It's a long thread, so I confess that I didn't read every comment.
Many digital thermostats require a battery in the thermostat. Have you checked, and if there is a battery, have you replaced it? The battery typically sustains the thermostat's memory and powers the switch.
Thermostats are essentially interchangable. I replaced the old mechanical thermostat in my rig with this
Honeywell. Two wires (for heat) and possibly 2 wires to control the AC (not usually there) and two screws to mount the thermostat. Someone already said it's a 5 minute job.
The act of turning the thermostat off then on will reset the safety interlocks in the furnace and allow it to run again until something in the furnace control board is not happy and shuts things down. It's possible that a thermocouple or similar in the control board is shutting things down.
Also, a dirty sail switch MIGHT be causing this. Normally, if the sail switch closes (turns on), there won't be problems keeping the furnace running. But it is at least possible that the sail switch is barely making contact, and after running a few minutes, the sail switch opens (turns offf) and the furnace shuts down. Not likely, but worth mentioning.
For $22, I'd throw a new thermostat at it and see what happens. If that doesn't fix it, then it's time to get serious and investigate what's going on with the control board and/or the sail switch. You won't regret having a nice, new digital thermostat from a manufacturer that specializes in these things.