It absolutely does. The model that came closest simply used that model twice in the same equation. One for the cup and one for the air.

It does? There is a fast drop followed by a long decay, exponential in fact. The cooling rate is proportional to the temperature difference, so the drop is sharpest at the very beginning when the object is hottest.

Are you sure? I believe Newtown's law of cooling says the temperature will drop sharply at the beginning:

dT/dt = -k(T_0 - T_room)

so T(t) = T_room + (T_0 - T_room) exp(-kt)

exp(-x) has a fast drop off then levels off.

probably dominated by the cup as the ambient temperature initially and then as air/the counter top as the ambient temperature on the longer time scale, once the cup and the liquid near equilibrium