In our tournaments, since all 4 players are opponents, all sign. It's kind of a pain in the a--, but there are reasons just like you've stated.
I've never really thought of the problem as you stated it (he signed a card with when you know he had the incorrect score). Usually, once the next tee shot is hit, the score is what it is. So this is a weird. How would you verify he signed the incorrect score? That's why our group always calls out their scores, so there is evidence (unless everybody's drunk).
If someone calls out an incorrect score, politely say, "I think you had a 10 on the hole" and recount the strokes. If you're not sure, it is a gentlemen's (and ladies) game and move on. But I would be counting his strokes on the remaining holes.
You're not being a prick. You are playing by the rules. So should he.
As for the talking, just say, "Talking during another's turn is a rules violation that could lead to DQ". Jokingly tell them "No charge for the rules lesson." They might get PO's at you, but what do you care. He's a cheater.