Does Facebook keep track of me? I wonder, since Facebook tracks all activity you do online when you're logged in using their like button and share buttons for example.

Because Facebook is integrated in iOS, does Apple allow Facebook to keep track of my Safari / Chrome searches because I'm logged in iOS? Or does it require a Safari/Chrome login specifically?

You don't have to know the exact or full answer, if you know a little detail (with source please) please add it as an answer.
No social network is fully integrated into iOS, only the login is. With iOS 6 Apple introduced the Social Framework which, as explained under the what's new page: 


  The Social framework provides a simple interface for accessing the user’s social media accounts. This framework supplants the Twitter framework that was introduced in iOS 5 and adds support for other social accounts, including Facebook and Sina’s Weibo service. Apps can use this framework to post status updates and images to a user’s account. This framework works with the Accounts framework to provide a single sign-on model for the user and to ensure that access to the user’s account is approved.


When accessing any social network's API (which means retrieving information, posting, etc.) from a different source than the actual social network, an OAuth process is perform to authorize the web page, app or otherwise which is making the call to access the social network's resources on behalf of the user. 

With the inclusion of the aforementioned framework, developers could acquire this authorization without having to perform the whole process, as the OS had already done it. Doing it like this, the developer just had to ask if there was an logged in account configured for the needed social network.

From a user's point of view, this inclusion meant that you didn't have to log in in every App which implemented social actions.

Consider too, that iOS Apps are sandboxed, which means that each App has its own space in disk, with its own directories, which act as the home for the app and its data, and no other App has direct access to these.

So, I don't believe that any social network has access to your information or movements, as the SO integration is just a way of easing the connection process with the required social network and not that the user is granting it access to any information.