Does an account has to be logged on to trigger the Time Machine backup, or does Time Machine run without any session active?
Time Machine is a system-level daemon, not a user-level process. Time Machine will work even if you are not logged into your Mac.
No. Mac OS X doesn't mount external drives so even if backupd was told to back up from an ssh session using
tmutil startbackup the external drive that Time Machine normally uses would be absent and the backup would fail.
It is possible for the disk to be mounted while all users are logged out, but MacOS and other backup programs do not incorporate this feature. The only one that can mount drive with no one logged in is CarbonCopyCloner. Of course, it's not as simple as time machine.