R – Flex web application control equivalent to mx.controls.HTML

There are a couple of ways, but neither actually have the PDF in the Flex App:

1. Convert the PDF to SWF. Use this tool or one like it to convert the file over.
2. Use HTMLComponent, a method that uses an iframe over your flash/flex to make it appear like an external page is in your app. There are a few downsides to this method however - most of them described in detail at Deitte.com.

I'm not sure what you mean by having the HTML login form reveal a Flex login form, but if what you're trying to do is establish communication between HTML/JavaScript (however rendered, PHP or otherwise) and Flex, then what you'll want to use is flash.external.ExternalInterface.

I recently posted a complete example here that illustrates how to set things up on both ends (HTML/JavaScript and Flex), and how to get them to talk to one another. You should be able to copy-paste both code snippets into a new Flex project to see a working example, and tweak as needed from there. Good luck!