Scoring is where most quizzes lose their goodwill. Not because hosts cheat, but because the rules were never stated and get invented halfway through, always in a way that seems to favour somebody.
Decide everything before round one
State the rules out loud at the start: one point per answer, how near-misses are treated, what happens on a tie. Two minutes of announcements prevents an hour of arguing. Write the rules down where people can see them. A rule that exists only in the host’s head is not a rule as far as a losing team is concerned.
Teams mark each other
Swapping sheets between tables halves the host’s workload and removes any question of favouritism. It also means every team hears the answers twice, which is where the learning happens. Read answers slowly and spell anything unusual. Most marking disputes come from a team not hearing the answer clearly rather than disagreeing with it.
Near-misses and deductions
Accept answers that are clearly the right thing spelled wrong, unless spelling was the point. Nobody came to be marked on orthography. Never deduct for wrong answers in a social quiz. Penalties teach people to write nothing, and a room full of blank sheets is not a better evening for anyone.
Keep a running total visible between rounds. A quiz where the standings are a mystery until the end loses the tension that makes the last round worth playing.