Is it possible to add something under each tumblr post, through tumblr theme, that could be visible through tumblr dashboard?
If yes, then how?
No, this is not possible given themes only affect the actual website of the Tumblr site. Themes cannot affect posts outside of the site that it's posted to. Themes purpose is to change how posted are viewed and presented on a site, not to change the content in each post.
If you want a signature or something on each post, you have to manually add it to each one as you create (or edit) the posts.