I want to know if this is possible with a Github App/
1. On PR Merge, the maintainer is asked a question, with three choices (whether it's a button or checkbox isn't important).
2. Their answer to that question is recorded.
And are there apps that do this kind of thing (adding form elements on specific action, with response recorded) that you recommend I look at?
It sounds like you're asking if you can add HTML form elements to GitHub's web interface by employing GitHub Apps. If that's the case, then no, that's not how GitHub Apps operate. However, you could have an automated conversation in the comments of a pull request with a maintainer or respond to "commands". For an example of a GitHub App that responds to commands you can take a look at the probot/reminders app.
I hope that helps!