Yes, you might want to do some research on it.

The core idea is that you aren’t using 1 unit of electrical energy to create 1 new unit of heat energy. You are using 1 unit of electrical energy to capture several units of pre-existing heat energy from the environment and pump it indoors.

So the term efficiency with heat pumps typically refers to COP Coefficient of Performance. It a measure of how many watts of heat you are able to pump per given watt of electricity used. So 400% would be a COP of 4.0 which is definitely possible with air source heat pumps in milder weather and ground source heat pumps in pretty much any weather.

If I can spend 1 joule to transfer two joules of enthalpy from outside to inside, then that is 200% efficiency.

Burning 1 joule of gas would give you 1 joule of heat in a perfect world. That is what a non pump does.

Where exactly does that first spent joule go, thermodynamically speaking? Is it also converted to heat within the electromechanicals of the heat pump?

Yes so not only are you capturing heat from the environment you are turning electricity into heat in the compressor, both are delivered as heat to the interior. This is an advantage compared to cooling where you must reject both the interior heat and the heat of compression.

To add to the other answer, if you calculate the absolute limit on performance for heating and cooling using the Carnot cycle, the 'heating' case is always going to be '1' higher than the cooling case, because you also get to use the 'work' heat.

More technically it would be a coefficient of performance of 4, which in some frames of reference is like 400% efficiency.

If you measure the efficiency of heat output relative to the electrical input power, then yes. The trick is that there's actually another input: latent heat in the environment. But given that this is abundant and freely available we don't usually bother to include that in our calculations.

Yes, because a heat pump is taking the heat from the outside environment and moving it into your house. You’re only paying the “transportation cost” and not the actual cost to generate heat. The heat was generated by the sun.