Re: How to move the admin bar functions into the template and remove the admin bar
You are perfectly able to move the adminbar, style it, create a positioning context of your choosing.
This is a styling issue and as such you should do this work on a child theme to preserve any changes you make to stylesheets and/or files.
As to how complicated this is is a question of your coding knowledge and BP/WP structure.
Giving any more detailed advice is difficult without having specifics to hand on what you wanted to do.
I’m puzzled by you remarking that you can live with the adminbar scrolling as this is not the default behavior, by default the adminbar is positioned ‘fixed’ i.e stays at the top of the browser viewport.
Actually taking the adminbar links and placing them elsewhere – in effect removing the adminbar is slightly more complex but doable but again you would need a good idea on where they would be going.