I am a developer who has never personally partaken in social media (in any form) for reasons completely outside the scope of this question. I am "off the grid" (no Facebook, Twitter, etc accounts). I'm currently building a web app and would like the app to have a presence on Facebook, and possibly even "port" my app over as a Facebook app.

My understanding of Facebook Apps is that they're just normal web apps that get <iframe>d into a Facebook page. The app is actually hosted on your server (not FB's servers). But this got me thinking:

1. Don't Facebook Apps have "profile pages"? Is there anything developers can do to customize the behavior of their own profile pages?
2. Do apps have the ability to do things like MySpace themes used to do (i.e., customize and interact with User profile pages, Groups, etc.)?
3. Do Facebook Apps gain any sort of extra capabilities (inside of Facebook) that a normal web app would not have? It seems to me like if all a Facebook App is, is an iframed-web app, that it would still need to communicate with Facebook via its many APIs, just like a normal app would have to, right?
4. If it's not possible to write an app that can customize the UI or behavior of user profiles and other pages, then how do games like "Farmville" interact with User profiles so that you see updates to profiles like "John Smith reached level 2 of Farmville"?

Basically, I'm asking any battle-worn Facebook app developers if my understanding of Facebook Apps is correct, or if I'm missing anything big here.

It's my understanding that for security reasons (obviously) Facebook doesn't allow apps to customize anything outside of the iframe it lives in. So if I want my app to appear like it's "interacting" with its Facebook users, it looks like I just need to publish stuff to the users' news feeds to try and encourage people to use my app (please correct me if I'm wrong here!).

Thanks in advance for any corrections, clarifications, advice or suggestions!