Well, we could see him avoiding the political pitfalls in France that John blindly went into IOTL, but he's not going to escape the usual issues of running a disparate hegemony in France, namely diverse and rebellious nobility, and Capetian suzerainty and meddling; while his authority in England wouldn't be undisputed (there's a reason why John's regency wasn't that challenged).
Now, Richard had a quite favourable military situation in Normandy, and without the succession crisis that happened after his death, I don't see Philippe II being able to break it in short term. Still, the war against Capetians and their French or Aquitain vassals was a real drain for Plantagenet's treasury, and as long French kings are able to pull a pressure by warring or supporting revolts (would it be only politically), Richard won't be that able to assert its authority (which is one of the reason of the truce of Gaillon) something that would probably continue up to his death (which, as
@Thoresby said, could happen sooner than later, his death in 1199 being representative of his behaviour : less because he was a fool, tough, than because undergoing trough the whole knightly charismatic behaviour was the best way to seal a disparate army and vassals).
Would Richard finally get around to siring an heir, or did he just not care? If not, would Richard try to set up Arthur as his heir (especially if Arthur is an adult)?
I wouldn't see him having an heir : Berenguela and Richard barely shared time together, and there's no real clue that it would change. If he does, it could arguably complexify things.
As for Arthur, Richard Oc e No wouldn't be that pressed to make his mind until he's on his deathbed, as usual : the problem wasn't only the age of Arthur, but as well the disunity of the Angevin demesne. Most of French nobles preferred Arthur to inherit at least Brittany, Anjou, Maine and Normandy; while Anglo-Normans lords rather elected John. Aquitains would probably end up supporting whoever let them alone.
In a situation where John and Arthur are still both alive, you would have a succession crisis, that could have as arbitle (who else) Philippe II trough his suzerainity over french holdings of Plantagenets : it could arguably end up differently than IOTL relatively easily, such as division of the Plantagenet ensemble but without real Capetian advance (altough them swallowing up Auvergne is more or less bound to happen IMO)