I’ll start by saying I’m not a physicist so I can’t say anything about relativistic effects, if that’s what you’re interesting.

In fluid mechanics we talk about “stagnation properties”. This can refer to a variety of fluid properties, commonly pressure and density, but most commonly temperature. Stagnation properties are defined as the value of a certain property when the fluid is brought to rest. For pressure and density this requires the fluid to be brought to rest isentropically (no entropy rise) and adiabatically (no heat addition or removal), but temperature only requires it to be adiabatic.

Using stagnation properties results in concepts of static and dynamic properties - the stagnation property is static + dynamic. A static property is the property that a fluid particle experiences when it is travelling at the same velocity as the fluid i.e. it is static relative to the fluids. For instance water in air will start to condense when the local static temperature and pressure fall to the dew point. A dynamic property is the component of the stagnation property due to the relative kinetic energy. This means that, yes, the temperature perceived by an object depends on the speed it is travelling at relative to the fluid. At high altitudes the temperature may be 230K, but an aircraft flying at high speeds experiences a stagnation temperature of maybe 260K (the exact value depends on the Mach number).

They usually have a thermoelectric module embedded in them, which generates electricity using a temperature difference over a semiconductor. The heat from the stove warms one side while the air flow from the fan cools the other. This is why they usually have some form of fin array on the top to make heat transfer to the air more efficient. The module drives an electric motor attached to the fan.

The same principle could also be applied with a Stirling engine - the temperature difference between parts of the engine can be used to produce mechanical work and drive the fan directly, without electricity.