Ciaira Castorena, the first situation you describe is expected behavior for a user like yourself with Full Access as an Admin because your access supersedes all other permissions. You can reference About System Administration Permission Levels and Core Help for more on this.
I don't know why the at mention of you personally in a direct message would not show up in your stream. I would expect that it would. Please don't hesitate to file a private support case for that particular point.
I've been testing this with a full access user as well and I received the inbox notification for both the DM's, one with the @mention of the space as well as an @mention for my user. Because of your permission level, I would expect your user to see both messages as well. Since this is not the case I would recommend creating a private support case. This will allow the support team to further review your permission and the site configuration.
Hope this helps.