Well, it is accepted that Lovecraft and Howard were acquainted. Howard wrote his Conan stories over a span of 4 years, and having Lovecrafts mythos, which he surely read, to inspire him, is evident. As to Conan living in the world of Cthulhu, just read the descriptions of creatures in Conan stories, read how scholars describe the evil gods and demons. They are described just as Lovecraft did.
The primary difference in Lovecraft and Howard is that in Lovecraft, you meet a monster and go crazy, it being unfathomable or indestructible. In Conan, you meet the monster and whack it, sending it back from where it came. Of course, you can do this in Lovecraft too, but most Lovecraft characters are scientists with little combat training.
I have always thought that Stygia was ruled by serpent people, it was always how I understood descriptions of it in Howard.
I would imagine that most fans of Conan and Cthulhu see the two linked, it being the same world, just at different times. I dont see why it cant be, in your game. It is in my game, but it is also more than just another time period in the mythos. Conans world is far more magical than Lovecrafts settings, so the gods are able to manifest easier. On the other hand, Conan sends many of them back to the great beyond. Here is a fair comparison of the two worlds=
http://kingsofthenight0.tripod.com/mythos.htm