1. That's the tough issue. Timing out the session and updating the database are going to be problematic.
2. You should be able to set and read cookies from both. Whether ASP can use the FormAuthenticationTicket or not is something I can't answer. The two issues with simply setting a cookie and reading that for logged in status are encryption/spoofing and clients accepting cookies. You can always expire a cookie to log someone out after a time period.
There is another option that might work, running the ASP app inside an IFRAME on an ASP.NET page. Not sure if your app would allow that or not.
Jeff