I know that the US Mint used to sell there old dies into the open market. Does anyone know if any 1893 S Morgan Dollar dies were among those that were sold?
In modern days, yes. But in the days of Morgans, not so Frank. I almost bought a Morgan obverse die (think it was 1880-P if memory serves) from Bob Cambell some years back. And it was not mutilated or defaced in any way. I've got pictures of it stored on an external drive. Only reason I didn't buy it in the end was the price tag - $11k.
No, and while I can't swear to it I'd bet the Chinese get them one of two ways. 1 - they create transfer dies from genuine coins. 2 - they carve the dies from scratch. My only point is that genuine Morgan dies, along with some others, do exist and they come onto the market every now and then.