Is there a way to share cookies across different domains and different applications (classic ASP and ASP.NET)
No, there isn't.
The issue is the cross domain one, rather than the asp.net/classic asp and is security reasons.
If the domains are subdomains you can share the cookies, provided you use a cookie path that both can access (ie. for the domain sub.example.com you can read and write cookies using the example.com domain).