Wouldn't that cause problems with things like cookies, browser detection scripts and so on?

It shouldn't if you are designing for it. That is a common piece of the MVC (Model, View, Controller) design model. You have a central controller that takes user input and displays the appropriate content based on input and state. All cookies are set with the path of your controller or just "/". Browser detection, etc isn't affected in anyway that I can think off.

It is not like you are creating a proxy, you are simply passing control to another php page temporarily and then you get control back so you can include a additional pages if necessary. My company has a standard jsp template (we are a java shop, but the premise is the same) that we use which dynamically includes headers, navigation elements and the body of the webpage based on database entries.

-Mike