1. You see the "edit" only because you are logged in. Otherwise is not there for the rest of the world :)
2. When you click an "archive link" (like month name, calendar) then WP uses the
archive.php (a.k.a. archive template) to display the posts. It has one Loop which will define how the posts are displayed = what template tags are used, including the call for Comments, no matter in which category they are.
2a. However you can have a different view if you click on a "category link", like a category name, and if your theme has a
category template (or more for different categories) then you can have a different display for that.
I know ate the beginning this whole "template thing" sounds confusing :)
My advice: open the Template Hierarchy in the Codex and in the same time follow my Visual Anatomy and you'll see better which template kicks in for different views :)