We had an established company incorporated in Australia. When we got into YC we set up the company in the US same as all the other YC companies, and transferred our IP to the US company.
Many YC companies already have products on the market.
No, there's no problem with a non-US resident owning a US corporation.
18 months on we still don't have visas as we haven't found it necessary to be based in the US - or not important enough to subject ourselves to the expense and distraction of the visa application process. We come and go from time to time on VWP. Only now the company's becoming more solid will we look at obtaining visas.
Many YC companies already have products on the market.