Facebook doesn't let others post directly to a page's timeline unless posting as the page (i.e. page admins). Posts by others show up in the visitor posts area buried in the left column.
Is there a way to have an app to post directly to the timeline? For example, could an app named ExampleApp post to ExamplePage and ExampleApp's post show as being posted by ExampleApp, not by ExamplePage?
No, that is not possible. You can either post "as User" (with a User Token and publish_actions), or "as Page" (with a Page Token and manage_pages/publish_pages).