Is it possible to have a web custom control( not a user control) in the app_code folder of a web project?

Or must it be inside a class library by itself? Do you compile the web site to get the dll then add that to the toolbar?

Yes you can find the info here:
http://alpascual.com/blog/al/archive/2006/06/19/The-power-of-CompositeControl.-Another-skeleton-class-.aspx
Cheers
Al
My Blog
MapStats.NET
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Sorry, I'm not following. What does extending a composite control illustrate about creating a custom web control inside the appcode folder of web project.