So, going to game 2 - we have a player who has failed to shuffle their graveyard from game 1 into their deck.

They present, opponent cuts, then they count 7 cards face-down on the table. *Before looking at any cards*, they notice they haven't shuffled their yard into their deck.

Here, I'm giving the penalty of TE-DP, and a warning - but, within the fix, do we instruct the player to mulligan an ‘opening hand’ they haven't seen, or will we shuffle the 7 unknown cards alongside the yard into their deck, and instructing them to draw 7 new cards?

If they haven't seen the cards then they're not a hand yet, they're just 7 cards in the library. They shouldn't need to go to 6.