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.

Riot is a replacement effect. Players can't respond to your choice of a +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. In the case of Arcbound Slasher, however, it's toughness will probably be 0, and it will be put into its owner's graveyard before it can attack.

If this creature gets enough -1/-1 counters put on it to cause it to go to the graveyard, modular will put a number of +1/+1 counters on the target artifact creature equal to the number of +1/+1 counters on this creature before it left the battlefield.