I have actually figured out the solution to this. What I did was create a redirect page that essentially grabs the user credentials from the initial login to the Sharepoint site then passes those credentials to the web application. The page viewer web part initiates the connection to the redirector.aspx (hosted on a separate machine or same machine, either way works) and because the credentials have been validated by Sharepoint the redirector simply grabs those credentials and passes them to the web application.
One thing to be aware of also is authentication between domains. If your web application is hosted on another machine but residing in the same domain then this way should work for you. If your web application is on another machine that isn't in the same domain but trusted in the domain that the Sharepoint server is sitting on then you should still be good to go because of transitive trust, but that's another animal.
Bottom line, if the machine you are hosting the new web application on is either a member of the same domain as the Sharepoint server or on a machine in another domain that is trusted by the domain hosting the Sharepoint server you should be good to go using this method. I have set this up in a testing environment to try the "different domain, same trusted environment" and the scenario worked. I personally have my web application on one web server in one building and Sharepoint on another server sitting 2500 miles away but on the same domain and this is working like a champ. I'll also add that the SQL Database that is the backend of the web application is sitting yet on another server in a totally different location than the web and Sharepoint servers and this is still working so there you have it.
What triggered the idea to do it this way was started with this article: http://social.msdn.microsoft.com/Forums/en-US/sharepointdevelopment/thread/3666ed41-3d9f-49cb-b3b6-9299dfc84848/
The article talks about using a "Content Editor WebPart" but if you pay closer attention you'll see that the idea pops out by one of the repsondants about using the "Response.Redirect" in a login_redirect.aspx page, which is exactly what I ended up using and it worked great and it solved issues 1 and 2 together.
To solve issue 3 the solution resided in changing permissions on the site to not inheret from parent and to set up individualized permissions for the site instead of using Active Directory Groups (e.g. BUILTIN\Administrators, DOMAIN\Domain Admins, DOMAIN\Domain Users, etc...) I removed these accounts and just added the users on an individual basis and Voilah, it worked!
Hope this helped and Good Luck!