Player A plays Emissary of Grudges and chooses Player B.
Player C steals Emissary of Grudges with Mind Control. Can Player C look at the named player? Can they activate the ability and reveal the name if Player B targets them with something?
I'm getting thrown off by the phrasing of "Reveal the name you chose:". 'You' should refer to the creature's controller, but Player C didn't choose a name.
Player C has no access to Player A's secret choice and cannot activate the ability. We can see this reflected in a ruling on Emissary of Grudges:
If you didn’t choose an opponent when Emissary of Grudges entered the battlefield, you can’t pay the cost to activate its last ability. This will usually happen because you gained control of another player’s Emissary of Grudges or because a creature already on the battlefield became a copy of Emissary of Grudges.
As you mention, Player C never made a choice, so they cannot reveal the choice they made. They also have no privilege to access Player A's secrets; they just control a creature.
We see this same ruling and behaviour on Stalking Leonin.
No, Player C cannot activate the ability or look at the named player.
From the rulings on the gatherer page of Emissary of Grudges:
If you didn’t choose an opponent when Emissary of Grudges entered the battlefield, you can’t pay the cost to activate its last ability. This will usually happen because you gained control of another player’s Emissary of Grudges or because a creature already on the battlefield became a copy of Emissary of Grudges.
