When a creature turns into a frog, does damage stay on it?
Damage stays on the creature.

This rule outlines all the situations in which damage would be removed from a creature, without a card explicitly saying so:


  119.6. Damage marked on a creature remains until the cleanup step, even if that permanent stops being a creature. If the total damage marked on a creature is greater than or equal to its toughness, that creature has been dealt lethal damage and is destroyed as a state-based action (see rule 704). All damage marked on a permanent is removed when it regenerates (see rule 701.13, "Regenerate") and during the cleanup step (see rule 514.2).


(emphasis added) Note that this rule says nothing about the creature turning into a different creature or changing its power/toughness or other characteristics. Since there is no rule that specifies that damage should be removed when those things happen, the damage stays.