Enabling remote login without password authentication

To disable password authentication you need to edit /etc/ssh/sshd_config. In Terminal:

sudo vim /etc/ssh/sshd_config

(or replace with your favourite editor)

Look for the line:

#ChallengeResponseAuthentication yes

And change it to:

ChallengeResponseAuthentication no

That is, remove the # at the beginning, and change yes to no.

All subsequent ssh logins will now REQUIRE you to use an ssh key and will not prompt for a password. Ensure your ssh key works before doing this if you have no local access! There's no need to restart the ssh daemon or anything because it is started on demand for each new incoming connection. Note that you may also see a PasswordAuthentication line, but that defaults to no already.

If your ssh keys aren't working then that's a separate issue really, but this answers the question as it is posed now :)

The only time the machine will need the filevault key to unlock is at boot, at all other times the system is effectively decrypted with the key in ram.

As soon as the machine powered off, it will be locked again.

You do not need to be logged in, and waking the machine up from sleep is no issue.