In answer to your concerns and simply speaking…
1) There is a difference between heat and temperature. A solid object has a higher ‘specific heat constant’ than a gas. As such, the solid has more heat, or energy, in it, for the missile to track. If you look at exhaust through IR, the exhaust vent glows A LOT more than the gas.
2) This is to do with heat transfer. Put simplistically, for a given amount of heat, which has to pass through metal to vent, passing it through a larger duct will ‘spread’ that energy out. Also, the metal duct itself is being likely being cooled by the external airflow, therefore helping the heat to be transferred to the atmosphere.
Hope it helps, if only a bit!