...A fundamental question: Is this electric heating rod in the air heat pump only there to assist when the air heat pump can no longer produce enough heat at particularly low outdoor temperatures, or is it also installed to protect the outdoor unit from frost damage?
The former is correct. The outdoor unit would first have to be defined. This can only be the evaporator (split systems) or the entire air heat pump. Air heat pumps are the most sensitive heat generators, as can be seen here, for example.
Precise planning/dimensioning of the
complete system is particularly important here. Significant mistakes have been made and are still being made, which have unjustly discredited the air heat pump and created a fear of the "evil heating rod". In well-planned systems, the proportion of the heating rod is barely < 1% of the annual heating workload! So a minor "cost" problem within the overall energy balance. In poorly or incorrectly planned systems combined with possibly wrong device selection, however, this can quickly become a money pit with excessive consumption costs.
...Couldn’t it—if it is not absolutely necessary for the protection of the system—catch the cold peaks instead of the heating cartridge of the air heat pump?
"Instead of" (physically) will probably make little sense! However, every kWh that is
actually contributed by the wood stove improves the energy balance, especially when it occurs at particularly low outdoor temperatures. The annual performance factor of the air heat pump improves. However, the expected effects must be assessed
beforehand. For example: Is the wood for the stove "free" or does the fuel have to be purchased? The JNG is fundamentally thought to be poor here! With a planned controlled residential ventilation with heat recovery, the overall balance can be improved by a sensible design of this.
...Can the air heat pump then be designed smaller?
I would strongly advise against this! Moreover, according to what criteria is the air heat pump designed? Annual maintenance costs essentially do not arise, although, in my opinion, totally nonsensical, maintenance contracts are also often sold here. Exceptions are systems where a refrigerant inspection (> 3 kg content) is required.
Best regards.