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 a creature entering the battlefield has riot but can’t have a +1/+1 counter put onto it, it gains haste.

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.

If a creature enters the battlefield with two instances of riot, you may choose to have it get two +1/+1 counters, one +1/+1 counter and haste, or two instances of haste. Multiple instances of haste on the same creature are redundant, but we’re not going to tell the Gruul how to live their lives.

Because it’s a loyalty ability, Domri’s first ability isn’t a mana ability. It can be activated only any time you could cast a sorcery. It uses the stack and can be responded to.