Does the pilot flame of a gas furnace stay on for the duration of a heating cycle?
This is an edited answer

After reviewing the additional information provided above along with relevant information obtained from this post: What is wrong with my gas furnace? I have entered my answer below.

The "something red hot" in your furnace is apparently a "hot surface ignitor".  The red hot glow is produced by passing current through the ignitor and is the intended function.



Now, to my concern for your safety:

I originally stated that there did not appear to be a thermocouple on your furnace.  While I maintain that there is indeed no thermocouple, I also add that the hot surface ignitor is likely monitored by the control module in the furnace.  While I am not intimately familiar with this furnace nor the application of such an ignitor, I can safely presume that the control module senses the resistance of the ignitor and can tell this way whether the pilot light is lit or not.

One way to verify my assumption about the control of gas flow would be to remove the connector for the ignitor while the unit is off and then see if you can hear the gas flowing (or can otherwise sense the gas flow - temperature, smell) when a demand for heat is made.  Just make sure you do this quickly and safely! (i.e. Do not reconnect the ignitor unless the thermostat is off!!!)