Conan's murderous indignation and fury would likely match my own intense hatred of self insert/bodyswap fanfiction.
Any way, he'd probably just tell Darry to go fuck himself at the first opportunity, murder all of Robert's assassins and likely send their corpses back with a note that reads "By Crom you fat fool! I desire only one thing and neither a crown nor your corrupt continent can grant it to me! Leave me in peace that I might pursue it or I shall come to Westeros and liberate your head from its bulbous body!"
And that one thing, by the way; is vengeance against whatever god was foolish enough to blight him with a new life where he's forced to deal with every trapping of civilization he holds nothing but deep abiding hatred for.
He'd bulk up, became a legendary sellsword and if he was absolutely compelled to return to Westeros it would certainly be in the employ of Stannis.
He'd probably tell Stannis to broom his ugly ass wife in exchange for Deanarys to legitimize himself and offer council of his own experiences..Also he'd eventually festoon the Red Keep with Melisandre's insides..the one thing Conan hated more than Demon apes and sodomy toads in the towers of demented sorcerers were life sucking thot style witches.
Which Mel is.
Stannis will probably agree after Conan the platinum sword points out he can always make Shireen lady of Storm's end and make her hand of the king so that she'd rule by her father's side until his next heir comes of age and then prolly stay on well into that reign if they get along well.
After that he'd try to fuck off or get Stannis to commission a national army with him at the head, so he didn't have to deal with politics.
Prolly get shitfaced with Tyrion who'd survive because he's a clever prick and I could see Conan arguing that a cobra in the skin of a lion is a fitting heir to Casterly Rock and no greater insult to the man who wronged the realm with his monstrous offspring.
What does he do to Khal Drogo as prophecy dictates that Dany will give birth to "The-Stallion-That-Mounts-The-World"?
Mock the shit out of the prophecy as he always does, but Deanarys would never get pawned off on a Khal because there's no way in hell Conan would want anything to do with Westeros unless forced too and then he'd just beat the fuck out of Khal Drogo and take the Khalassar, unite all the Khalassars and conquer Volantis to use their resources and navy...and again he'd probably just show up at White Harbor or Dragonstone and be like "Fool Stark, take my legions, my riders and my marines and settle those blond imbeciles..also take my sister and make a decent husband to her" or "Stannis, you have more sense than your brother abandon this murderous folly against me that I might aid you in ripping that incest spawned tabby what fancies himself a lion off his stoop so that I may have a measure of peace in this benighted world"
He'd try to nope the fuck out of confrontation with White Walkers as his default response to magic is "Hahah fuuuuccckkk this shit!" and when they make that impossible they find out..the indignation and disgust of a Cimmerian is more deadly than all their legions of doom and dark magic combined.