Electronic – the reason for a fuse next to a thermostat

Firstly think about what the purpose of the fuse is. Basically the thing you don't want is for whatever is being protected by the fuse - components, wiring, connectors - to go up in flames before the fuse breaks under fault conditions (either single or double depending on how safe you want your design to be). Remember that when the fault occurs the system is already damaged and the fuse is only there to prevent a dangerous failure.

So lets analyse the failure paths in your circuit:

1. the wiring or connectors between the fuse and the unit shorting out.
2. D1 failing open or closed circuit.
3. 10ohm resistor failing open or closed circuit.
4. D2, capacitor or regulator failing open or closed circuit.

In scenario (1) the fuse just needs to blow before the wiring does, so unless the wires are particularly thin even a very high value would be fine. In (2) and (3) there is no short circuit risk. Scenario (4) is then where you start to see a problem. If any of these fail closed circuit then current will flow in a loop through D1 and the 10ohm resistor. This will put more than 10W through your resistor which exceeds its rating. Even so it may not pose a dangerous risk (the resistor will likely just burn out and either short - which will then blow the fuse, or open circuit), but you might want to either reduce the fuse to a size that protects for this or get a non-flammable fusing resistor (which you probably need for the load dump rating anyway). As you identified you may not be able to reduce the fuse enough due to the inrush/load dump current anyway.

As an aside, note that the battery voltage can easily fall to 7V or less during cold cranking conditions. Your circuit as it stands won't provide enough margin to prevent the regulator dropping out so you might need to reconsider your design if this is a problem.

Also keep in mind that most modern vehicles have a central load dump surge suppressor on the alternator so individual modules don't need to protect against this. This makes achieving cold cranking voltage operation much simpler.

Yes, the thermostat contacts control the current to relays within the heating/cooling system. I believe the current is on the order of 100 mA.