If Rhythm of the Wild leaves the battlefield at the same time that a nontoken creature enters the battlefield (most likely because that creature has a replacement effect, such as that of Rescuer Sphinx), that creature still gets a +1/+1 counter or haste.
If a creature entering the battlefield has riot but can't have a +1/+1 counter put onto it, it gains haste.
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.
Magic: The Gathering · ™ & © Wizards of the Coast · Illus. Tomasz Jedruszek
Card data via Scryfall. Rarix is fan-made; not endorsed by the rights holders.
A noncreature card that happens to be entering the battlefield as a creature will have riot (for example, Rusted Relic while you control three other artifacts). Similarly, a creature card entering the battlefield as a noncreature permanent won't have riot (for example, Thassa, God of the Sea while your other permanents contribute only four to your devotion to blue).
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.
A spell or ability that counters spells can still target a creature spell you control. When that spell or ability resolves, the creature spell won't be countered, but any additional effects of that spell or ability will still happen.
If a nontoken, noncreature permanent becomes a creature after it's already on the battlefield, it will have riot but it will be too late for the replacement effect to have any effect.
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.
Once a creature with riot has entered the battlefield, it keeps its +1/+1 counter or haste even if it loses riot.
