A: Yes, it's definitely possible. Usually SharePoint is setup with Integrated authentication in IIS. If your web app uses the same login/password to access the app, then these credentials can be "passed through" to the app from SharePoint. Just setup your app virtual directory to have integrated security.
Also, here are some things to consider:
1. Do you want the "application" to run within a SharePoint page? Maybe consider writing a .NET web part then.
2. Do you want the "application to run within the context of SharePoint (access to the SharePoint object model)? Then you have to put the app in a specific vdir location (_layouts folder) in order for it to work with SharePoint. This is documented in the SharePoint SDK.
3. Is the app running on the same server as SharePoint? If so, then you either need to exclude the virtual directory for the app from the managed paths or create a new web site and put it there.
Thanks to Nate B. for his help with this issue.