You have to take the damage within 5 seconds.
I don't think a tank would receive enough damage quickly enough from Dragon or Baron to activate Sterak's Gage. While they deal a significant smount of damage over time, they don't exactly deal damage in bursts; they deal it over a significant amount of time.
The fountain deals pure damage (which is different than true damage). Since only the fountain laser deals pure damage and pure damage bypasses the effects of many abilities and items (i.e. It will kill people through Tryndamere's ult, Kayle's ult, Kindred's ult, etc), it's quite possible that Riot decided that Sterak's Gage shouldn't activate from pure damage.