first question is did you even read the feat?
Divine Might [Divine]
You can channel energy to increase the damage you
deal in combat.
Prerequisite:Ability to turn or rebuke undead,Cha
13+,Str 13+,Power Attack.
Benefit: Spend one of your turn/rebuke undead
attempts to add your Charisma bonus to your weapon
damage for a number of rounds equal to your
Charisma bonus.
so yes you get the damage bonus for the entire round which includes multiple attacks if you got them and yes you can spend your turn attempts from extra turning for this feat since the feat adds to your pool of turn undead attempts
But do you really need to. Think about it, if your taking this feat you probably have a high cha which means a min of 4-5 rounds the feat is active and that also means around 7-8 turning attempts a day just from your cleric class. Most battles i've been in only last 4-5 rounds.