The history of news headlines currently available through Eikon Data APIs only goes back 15 months. But otherwise I don't see a problem with retrieving 10 headlines and stories per day for a company, provided there are that many headlines and stories for a company on a given day.
The starting point for the mouse pointer in my screenshot is the top left corner of the first right-click menu. You can also obtain the same commands from the hamburger menu button on the right hand side of the app title bar (just to the left of the minimize/maximize/close window buttons). But you need to be in the headlines view of the app, which is the default view the app opens with. In the screenshot you provided I cannot see the full app. I assume you're probably not in the main headlines view and are perhaps looking at the related headlines in the story view? If you include the screenshot of the full app window, that would be very helpful.