Conan doesn't strike me as a Barbarian really, let alone a Thief/Rogue. I mean this is only going by the movies, I'm not familiar with any of the other material, but... Just because Arnie sometimes puts on a bit of paint and mostly wears medieval porn star outfits light armor doesn't make him a Barbarian. He never really rages at all, he makes those stupid Arnie sounds but he doesn't actually go into a rage.
He also uses a wide variety of weapons, uses strategy sometimes such as the fight against those guys on horseback, he goes for trips and stuff, he uses shields sometimes.
He strikes me as a typical Fighter more than anything else. Meh, I could be wrong.
First, go read the books. They're great fun.
Second, Howard's original Conan isn't really that much similar to the Arnie's movies Conan.
Third in Howard short stories, he does "multiclass" a lot. From a "pure" barbarian, to thief, to a pirate, to a mounted warrior (with some archery thrown in), to a chieftain, a leader of infantry, to a commander commanding from the back row until the last moment, to a heavy armour horseman and a king leading his knights' charge into enemy ranks.