I beg you, if you are a developer of an open source app or program - add screenshots of your app to the README file. When looking for the perfect app, I had to install dozens of them just to see what the user interface looked like and whether it suits me. This will allow users to decide if the app they choose will suit them… Please, don’t think about it, just do it…
Yes. Git can store binary files fine. It’s not the most efficient for storing them, but it works, especially for a small number of screenshots. For updating and theme, that’s entirely up to you. It’s all a judgement call. If you want to show off your functionality (like a dark mode), I encourage you to include screenshots of it. If you substantially change your UI, update the images.
You don’t have to update for every new button you add. It’s more about giving a general impression of the UI. Is it minimalist? Is it a chaotic mess? Does it look like it fits in naturally with whatever OS appears to have been used? Does it look like any thought was put into UI and UX? Those are the kinds of things you’re trying to answer.