Will I get trophies for clearing lower difficulties if I play on a higher difficulty?
No, they don't appear to stack. You'll have to clear each difficulty level in order to unlock the corresponding trophy.
According to this trophy guide, on your successive playthroughs you can skip most of the levels - once you've beaten the game on any difficulty, you can change the difficulty level and replay just the last level in order to get credit for clearing each additional difficulty.