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Channel Smite doesn't define the type of damage done (it doesn't say it's positive energy damage, or negative energy damage) It simply says the weapon does more damage - so more Slashing, Piercing, or Bludgeoning damage. DR would still apply.
No it doesn't.
Benefit: Before you make a melee attack roll, you can choose to spend one use of your channel energy ability as a swift action. If you channel positive energy and you hit an undead creature, that creature takes an amount of additional damage equal to the damage dealt by your channel positive energy ability. If you channel negative energy and you hit a living creature, that creature takes an amount of additional damage equal to the damage dealt by your channel negative energy ability. Your target can make a Will save, as normal, to halve this additional damage. If your attack misses, the channel energy ability is still expended with no effect.
The creature just takes additional damage. It isn't an increase to the base weapons damage, just additional (but separate) damage.
You are spending a channel attempt (just like a normal channel), the damage only applies to certain targets (just like a normal channel), and there is a saving throw (just like a normal channel). If anything, it works exactly like a normal channel with the exception it only targets one creature that you hit with a weapon instead of a burst.