Hi,
1. I think you skipped the step to add the class name to the posts widget. That should do the trick.
2. Oh, I understand it now. Well, not via settings for sure. The current single page ID is sent to the search, so it would be possible to recognize where the search was origianlly initiated – so it is not impossible. But definitely a lot of trial and error custom coding for the query arguments.
We have a ready made solution for something similar here. This code restricts the search to the same category as the current post is in. In your case line 3 should be edited, by adding the relevant taxonomy name: $taxonomies = 'category, post_tag';