R: After your opponent places one or more cards to their discard pile do to your card effects, the take two damage if this card is face up in your momentum.

For this response does it include discarding their momentum, destroying assets/foundations, discarding from hand/top of deck, discarding cards from their card pool?

The way I interpret it is all of the above.

Also what about forcing them to recheck? That is your card making them add a card to their discard pile right?

Everything you listed except the re-check, because making a control check is a game mechanic.