Register/Login button on front end

There is a Meta widget which has the register and login links.
There is also a function you could put in your theme template file instead.
https://developer.wordpress.org/reference/functions/wp_loginout/

There are plugins to customize the login page. (It is not handled by the theme.)

Most themes show the Edit link to logged in users. If yours doesn’t, it’s pretty easy to add (or switch to a theme that has it).
https://developer.wordpress.org/reference/functions/edit_link/