The date on restart was current, so does that rule out the battery?
If you turned it off and on and the date remained current then yes the battery is fine.
Perhaps try an external keyboard and see if that addresses the keystroke issues. Any chance you have a computer virus? If not a new disk then perhaps a low level disk reformat (after backing up your work) would be in order but your backups may also be infected.
Agree, it is an unusual problem as normally the Windows installer will examine your system to confirm an install is possible, and inform you if not in most common cases.