Riot is a replacement effect. Players can’t respond to your choice of +1/+1 counter or haste, and they can’t take actions while the creature is on the battlefield without one or the other.

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.

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 another effect (or effects) modifies how much damage your creature with a +1/+1 counter on it would deal—by preventing some of it, for example—the player being dealt damage or the controller of the permanent being dealt damage chooses the order in which any such effects (including Uncivil Unrest’s) apply. If all of the damage is prevented, Uncivil Unrest’s effect no longer applies.

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.