Hi Yuri,1) remove the G wire on the circuit board ( board ) on the G terminal and put some tape on the end of it. Do not use the AC with it off. Wait a day or two and if the fan starts you may have a problem with the board.
2) If it runs with the G wire off then tap the board with the butt end of a big screwdriver in several places and if the fan stops then the fan relay in the board is faulty and shorting on. That is a older furnace ( 15-20 yrs old ) and the board may be dying from old age/have a problem.
I followed your instruction and had the observation and test below:
* Before disconnecting the Green cable ( see the attached picture#1 )*
10/12, SYSTEM=OFF, FAN=AUTO:
I heard the very short(<1 second) buzzing sound(not sure if "buzzing" is the right word, but the sound was just like a motor tried to start but immediately stop, or like the sound when sump pump starts) from the attic, where the furnace is located, for about 5~7 times a day(mostly in the early morning or late evening as I noticed, but I still couldn't find the pattern), it sounded to me it tried to start the motor for 2~3 times within a half minute period, then stopped trying. This has been an ongoing issue for several weeks.
10/13 morning, SYSTEM=HEAT, FAN=AUTO:
My wife turned the HEAT on, after reaching the target temperature, the FAN kept running, while the air blew out no longer warm, so I assumed the furnace turned the fire off. I waited for >40 minutes, the FAN was still running. Then I forced to set SYSTEM=OFF. After about 5~6 minutes, the fan finally stopped.
** After disconnecting the Green cable ( see the attached picture#2 )*
10/13 noon, I went up to the attic, disconnected the Green cable from the board, and left SYSTEM=OFF, FAN=AUTO
It has been more than 24 hours since I disconnected the Green cable(and kept SYSTEM=OFF), I haven't heard any sound from the attic yet, but I will wait for another two days.
When I was at the attic, I noticed another Green cable detached, I am not sure what's that for. I attached the picture below, can you advice?
Also, I want to mention again another test I did before(which I mentioned in my original post by following the Pepco staff's advice): I disconnected the front part of the thermostat(the main part of the thermostat, with circuit board, LED screen and control buttons…) from the wall part(where the color cables are connected), the problem persisted: either the fan ran for 30+ minutes then stopped, or I heard the short(<1 second) buzzing/starting sound from the attic.
So I don't think the problem is because of a bad thermostat control board, but I don't know about if something went bad with the connector part of the thermostat, or something wrong with the green cable(from the thermostat to the control board in the attic…
The symptom of the problem sounds to me like this: when SYSTEM=OFF or (SYSTEM=ON but FAN shouldn't be running), the fan tries to start(when I could hear those short buzzing sounds), when it fails to start, it may try again hours later. When it can successfully start, it runs for 30~40 minutes then stops)