Why does the pilot burn perfect before igniting the burners, then stutter while burners are on and go out after burners shut off?

SOURCE: Coleman Propane Furnace Good pilot light but feed/ignite main burner

If you have a volt ohm meter take the two leads off the gas valve and ohm out the connection should have resistance for the coil. You said you have 12 volts from the thermostat white and red in the heating mode you should have minimum 26 volts. If you want to check the transformer should have 28 volts. Almost sounding like the transformer is getting weak. You can get parts for a 20 year old unit at any HVAC parts house. Try Washer Specialties here in Wichita Kansas they are on the web. Lowes even carries Transformers. Has to be rated 120 volt primary side secondary side 24 volts and 40 VAC.

SOURCE: Laars boiler pilot light is blown out when burners shut off

Time for a cleaning and gas pressure test.

SOURCE: pilot light won't stay lit.

most common cause of the described symptom overheating the thermocouple! a thermocouple produces millivolts of electricity based upon a differential of temperature between the tip and the base. If it gets too hot it will not work correctly. If the pilot flame is too large or the main burner flame is contacting the thermocouple or the thermocouple is too cheap or old all can cause this issue.

SOURCE: cozy gravity one sided wall heater...pilot won't

how by extending the duct work will it make the gas valve work ?

if the pilotkeeps going out clean pilot jet , or if the power unit has gone in the gas valve then you would have to change the valve , don't get the flue bit ?

SOURCE: I have a GE (Rheem) SmartWater SmartShield Sensor gas cut off

The click you hear is the gas valve shutting off flow. It does that because the thermocouple (a rod sticking into the flame of the pilot) voltage drops, usually below 13 millivolts or so. Might be a loose connection from thermo to valve body (should be finger tight plus 1/4 turn) or a bad thermo ($13 at Lowe's). Good Luck.