Profle Icon with Vertical Tabs

Hi,
at the moment, there is no way to either move vertical tabs or the profile icon to somewhere else.
but feel free to suggest it using the feedback button on Edge (Shortcuts: Alt + shift + i )

in the meantime, I could suggest to stick tabs side by side.
by default, in Windows 10, you could stick 2 tabs side by side to fill the entire screen, or you could stick them side by side but only take half of the screen (either the upper half or the lower half), the 3rd way is to have 4 windows in a 4x4 fashion, which will take the entire screen space.
for more custom organizing you could use PowerToys.

Another thing you could do is to install your Email client as an app on Edge so you will receive native push notifications as well as notification badges on the app icon on taskbar.

press Edge's (...) menu button => Apps => Install this site as an app

hope that helps