Dunerider Outlaw's triggered ability checks to see if it dealt damage during the turn to an opponent of its current controller. If it dealt damage to player A, then player A takes control of it, the ability doesn't trigger.

If Dunerider Outlaw dealt damage to an opponent who later left the game before that turn's end step, its ability still gives it a +1/+1 counter.

Dunerider Outlaw's triggered ability gives it at most one +1/+1 counter each turn, regardless of how many opponents were dealt damage beyond the first.