But "the next person" also has a 1 in 6 chance they won't have to go at all because Holly might get the egg
Holly = 1 in 6 chance = 16.6666%
Next person = 5 in 6 times (Holly not getting the bad egg) 1 in 5 = 16.6666%
At each stage the next person has a higher chance of the getting the bad egg, but this is offset by the same magnitude due to the fact the person before might end the game before you even get a go.
They all have a 16.6666% chance at the start of the game, so going first is no advantage.
No, as the game resets each time.