1 : move this button (connections)
You'd need CB Auto Actions to insert a new menu item on the menubar. The connections menu item is also dynamic so this may not work out well.
2 : delete this button
For mobile, you can't. For desktop, don't set any tabs in that "Canvas Main Middle" position to display mode "Nested Tab Menu"; change them to "Tab Menu".
3 : delete this block (not just the css but all this block)
You can't. Canvas layout is designed for it. You'll need to use the other tab positions to customize a different profile view or you'll have to rewrite the canvas CSS to fit whatever layout design you're wanting. See the CB Primer Book for a tab position map. You can download CB Primer Book below.
www.joomlapolis.com/downloads/documents