Which file do I need to edit the All Posts page in the admin area?

You do not need to and absolutely should not be editing a core WordPress file to achieve this. There are almost certainly hooks for whatever it is you are looking to accomplish. You can take a look at questions regarding custom columns (such as Add column to pages table) or, if you’ll excuse my linking to something from myself, I have some slides regarding that same topic: http://slides.helenhousandi.com/wcphx2012.html#13

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