Would it be correct to say that before the round can be officially 'over' in Phase 10, everyone playing should be able to take a turn? For example: three people are playing Phase 10 and just started a new phase, we'll say Phase 6: 1 run of 9. Player 1 picks up from the deck and discards a card from his hand; Player 2 picks up that card and is able to lay down 10 cards and discard the last card-thus, completing and finishing the phase. However, would Player 3 be entitled to a turn before the round is completed? I can't recall if this is an actual rule in the Phase 10 rule book or if over the years our family had just adopted the rule to let the example Player 3 have a turn to try and be fair.
There is nothing in the rules about this. Yes, a round could end before a player gets a chance to play.
This isn't really any less fair than anytime a player other than the last in turn order ends up going out; in that case the last player has less turns than anyone else. Having 0 turns instead of 1 isn't all that different from having 4 turns instead of 5. So if you wanted a house rule to make it more fair, you should always end with the player who played last; not only if it happens in the first round.
The extra advantage of being the start player is balanced out by having the start player rotate.