Developer of The Ninja's Garden here, Maybe didn't have enough time to explicitly state/show in game, but you can destroy the obstacles :), thanks for playing anyway.
Well, some of the obstacles can be destroyed, how would I get past here:
without dumping into the hole and taking damage? Also there is a room with 3 cogs that either require some mad precision or 1 heart of damage to get through.
Yep, you either take 1/2 heart of damage, or you pick up the double-jump upgrade first. All depends on your route. You can also go for it last and use it as a deathwarp.