Uncivil Unrest

The damage is dealt by the same source as the original source of damage. The doubled damage isn’t dealt by Uncivil Unrest unless it was somehow the original source of damage.

If damage dealt by a creature you control with a +1/+1 counter on it is being divided or assigned among multiple permanents and/or players, that damage is divided or assigned before doubling. For example, if you attack with a 5/5 creature with trample and it’s blocked by a 2/2 creature, you can assign 2 damage to the blocker and 3 damage to the defending player. Those amounts are then doubled to 4 and 6, respectively.

If you choose for the creature to gain haste, it gains haste indefinitely. It won’t lose it as the turn ends or as another player gains control of it.