If a player deserves a second yellow should play stop immediately or can advantage be played and then the player sent off later?
No. Advantage should not be played for an offence which will result in a red card, regardless of what the reason for that red card is, unless there is an immediate goalscoring opportunity for the opponent of the offending player.
See Law 12, part 3 of the IFAB Law Of The Game. Emphasis is added.
Advantage should not be applied in situations involving serious foul play, violent conduct or a second cautionable offence unless there is a clear opportunity to score a goal. The referee must send off the player when the ball is next out of play, but if the player plays the ball or challenges/interferes with an opponent, the referee will stop play, send off the player and restart with an indirect free kick, unless the player committed a more serious offence.
Yes, sometimes referee could let the advantage and the game continue. After the next stoppage (and for a consequent fault only), the referee gives second yellow card without whistling the fault that caused the yellow. Example in this video at 2 min 18