TDP is not the power usage of the card. It is only a measure of how much heat is expelled. 105 W TDP is the same amount of heat as 3 soldering irons.
So yes, you do need the adapter.
Yes it is. The exact definition of TDP varies by manufacturer but it is usually either the peak power consumption or the nominal power consumption.
Neglecting the negligible amount of power that is transferred from the GPU to the display over the signalling cable all the rest must go somewhere and since GPU's don't make sound or move the only logical explanation is that it gets dissipated as waste heat.