My goal is to use a web app to bring up a Sharepoint .aspx page. When
I currently try to bring up a Sharepoint page through another
application it asks for a login.

There is certain code that you can use to take default credentials or
use impersonation for Sharepoint web services. Is there some way to
do this (use default credentials) when displaying a Sharepoint page
from another application?

I also tired taking a web part from Sharepoint (by exporting it) and
loading into an ASP.NET web application and I couldn't get it to work.
Is it possible to take a web part from Sharepoint and use it somewhere
else?