Is it legal to lay out your phase and play out your hand (placing remaining single card in the discard pile) in a given round before all players have had a chance to play in that round (i.e. pick up a card and discard)?
Short answer: Yes.
This is one of the main tactics in Phase 10. It is used to surprise the other players so they don't make their phases and only you move on to the next Phase in the next hand. Another benefit is you catch your opponents with many cards in their hands.
Why is it legal? Lets grab the rule book:
GOING OUT / FINISHING A HAND: After laying down a Phase, players try to “go out” as soon as possible. To go out, you must get rid of all of your cards, by discarding or hitting on an existing Phase. The player to go out first wins the hand. The winner of the hand, and any other players who also complete their Phase, will advance to the next Phase for the next hand. Players total the cards left in their hands. (The fewer cards left in your hand, the better!) All the cards are then shuffled and a new hand begins. (Remember, if you did not complete the Phase before another player went out, you must work on the same Phase again in the next hand.)
It's actually less about what the rules state, and more about what the rules don't forbid. They don't forbid going out immediately after making a Phase, in fact the rules practically encourage it: "players try to go out as soon as possible."