Yes, many problems it sounds like, and none of them seems familiar. No, not a typical experience.
Can you please try uninstalling and reinstalling the app (again)?
No icon to send mail sounds particularly interesting, it is supposed to be there, next to the "three dots" icon top right (on the compose screen).
Icon to the right of account still is there, on tablets and phones both, on the app's main screen.
Are you using a custom ROM by any chance?