Database connections are just connections. You can connect to the same database from any application that supports it without problems.
ASP and ASP.net can both use COM components without problems apart from the usual DLL hell and possible .NET interoperability difficulties.
You may experience problems with cookies because of the way application roots are written in cookies. A ASP.NET app will always have a virtual directory application (whether it's the server root or an actual virtual). ASP doesn't always follow the same pattern.
asa is for classic ASP. asax is .NET. The two don't mix.
Passing values between application contexts requires writing the values to a persistent medium (i.e. database, file, etc) or passing over the query string. As mentioned you can't share values using the *native* session technologies. That is not to say that a custom built session management system can't. However, you still need a way to carry a session identifier between the applications (usually querystring).