Regarding the following:
- Emailing Me with Feedback
- Sharing the App on Facebook/Twitter
- Completing a Tutorial
You can absolutely create GameCenter achievements for these. I have done similar things in many games (ie. we always have an achievement for competing the first level (which contains a tutorial) and for sending a "Challenge a Friend" email). Apple has never blinked at it. You're perfectly welcome to make whatever achievements you like, as long as they don't promote behavior that Apple otherwise frowns upon.
Regarding Unlocking Levels Through In-App Purchases
, you can, but I would recommend against it. A lot of users take pride in completing every achievement available in a game, and there's a gaming personality that relies heavily on things like this. To prevent these users from earning every achievement without an In-App Purchase alienates a good portion of your userbase, which is no way to get a successful app. But as I said-you can, if you want.
Regarding Rating the App on the App Store
, you can sort-of do this. You can have a button that asks the user to rate the app and takes them to the app store, but you can't get any confirmation that they actually did rate the app once they got to that page. So you'd have to just give them the achievement on the honor system, once they clicked the button, and hope they really did rate it. (Just like the "emailing you with feedback" one--you can guarantee they sent an email, but not that it contained anything worth reading (or even that they ended up sending it to YOU!)).
Hope this helps! Let me know if you have any other questions.