When my opponent counters cauldron dance, do they know what card im going to put from my hand onto the battlefield?
I know the will know the graveyard card.
In general, no. You choose whether to put a creature card from your hand onto the battlefield when Cauldron Dance resolves, not when you cast it (review C.R. 601.2 for choices made while casting a spell). If the spell is countered, none of its effects happen (C.R. 701.5a), so that in general, no other player will know whether you would have chosen to put a creature card from your hand onto the battlefield this way.
I know the will know the graveyard card.
A similar principle applies to Aether Vial's activated ability and Kari Zev's Expertise.