Does the power drop on a diode become entirely heat?
It's heat.
Unless your diodes are LED, where 10-30% of the energy goes out as light. The rest is heat, as usual.
Diodes don't store energy. At least, not intentionally and not in the meaning usually accepted in electronics.
The diode may have some parasitic capacitance, inductance or even a parasitic Peltier effect that can, in some cases, return some energy back to the circuit. The capacitance may even not be considered parasitic, as in varicap diodes, but they are a corner case.
Elements that are expected to store (and return) energy:

capacitors
inductors
batteries