Typically Japanese homes are laid out economically so the bath is right where the hot water heater is. In our house from the 70's it's probably not more than meter through the wall to the water heater. Even in apartment buildings, typically each apartment has it's own tankless gas water heater on the outside wall of the building so it's easy to meter usage separately.

That is what I always see in wet saunas/bathrooms around me. The bathing room is a main feature so things like main plumbing and water heaters are all placed around that to keep it simple. Recirculation additions after that would then be two simple connections off the back of the tub, probably near where the faucet goes.

Technically, you could also just pull refrigerant lines from the heat pump to the bath tub - you probably already run those to the indoor AC units you have in different rooms anyway - or connect the bath tub to the hot water lines you run to the radiators. Both imply a grey water heat exchanger built into the tub.

Depends on setup, of course. If there's a central AC blowing hot/cold air through ducts, you're stuck using a grey water loop. But it is the least attractive setup...