Unless I'm mistaken, it seemed to me that Froggy G's splash dash with one upgrade was sufficient to kill an enemy droid (creep) at the start of the game. Later, I used a fully upgraded dash against one and it still had some health left.
Do droids get more maximum health later in the match? If so, is this dependent on time, player levels, or something else?
As of Patch 2.13 (released December 2, 2015), droids no longer automatically gain health over the course of the match. Instead, droids gain health and damage through the Experience System. Each time a team levels up, that team's droids gain +2% damage and +3.5% health from their base values. A team gains experience when players kill neutral or enemy units or structures.
Edit: This answer is no longer accurate
Yes, droids gain more health as the match progresses.
In the recently released 1.2 patch notes (link), there is this entry:
Droids, improved base health and health increase during gameplay
From this, we can see that not only do droids gain more health as the match progresses, but after the 1.2 patch this will occur to an even greater extent than it did before.
