Hey there,

I am really sorry for the delay of this reply as it is a public holiday for us here.

Could you please explain to me how the red and green dots work on peoples names?

If the user doesn't log it out, this user will stay online mode until his user login session expired.
This is why i always encourage admin do not set too night for the user session lifetime (http://take.ms/9OXpS), if those user would like to keep them login when open the browser again, they can tick "Keep me logged in" checkbox before they login, so it will always appear user logged in when they close the browser then re-open again.

And how the 'sorting by recently logged in' works, as it is very vague and inconsistent.

Basically when the user click on this filter 'Sort by Recently Logged In' (http://take.ms/vkOeI) under member listing page, the system will always check on site regarding those existing user when is their last login date time.

For example :

User A - This user logged in on the site at 3pm.
User B - This user logged in on the site at 4pm.
User C - This user logged in on the site at 5pm.

If they never login again, this recently logged in sorting result will always show User C, User B, User A.
If User A login on the site today, so the sorting result will become UserA, UserC, UserB.

Is there a way to turn off the red and green dots/online user option?

You can disable this from your site backend > Easysocial > setting > user > layout > Display Online Status - NO