Never heard of this related to temperature, but this sounds like an issue with the go-pedal. Remember, this is a drive-by-wire truck; the pedal is just a big "volume control" variable resister. It's often called the Throttle Position Sensor (TPS), but it's really a Pedal Position Sensor. There's also a switch that opens and closes at the idle position. If you're getting absolutely no response from the pedal, that suggests that either the idle switch (IVS) or position sensor (TPS/PPS). You might find a complete replacement pedal assembly, with both electric components included, at a junkyard or the classifieds section of this or one of the other Powerstroke sites.
'Course, this doesn't explain the brake warning lights. When this happens, do you happen to notice the speedometer jumping up or down?
Just so we're all on the same page, does this happen even in Park (if automatic) or Neutral?
BTW, the red "BRAKE (!)" light is NOT just an "e-brake" light. That's a rather minor use of the light. It's major purpose is to light up when you have a brake hydraulic issue, like low brake fluid level or pressure. IOW, if it lights up while you're underway, you may have no brakes.