If I remove an app in Application Settings, does it remove my account and any of my progress / information in that app?
From what I understand, developers do not have to remove any data from the database unless specifically asked by the user or Facebook. So it really depends on the application and you should follow up with the application in question.
Not necessarily. Unless the app developer has designed his app to automatically remove your account information based on your removal from within Facebook, the app may still contain any information you already put into it. You might want to read the app's policy (if this even exists) on that as each will have their own way of handling it.
Regarding Facebook's policy on the matter, they are only required to remove the information which was provided from Facebook (your profile info, your pictures, your related info). Data you provided directly to the application is free to use how the developer chooses.
An article by ReadWriteWeb details some additional info to be concerned about when dealing with Facebook Apps.