There's no stipulation in the Find Familiar spell that states your familiar disappears when your character dies, or is knocked out. There's also this small statement within the spell about what a familiar does of its own volition:
Your familiar acts independently of you, but it always obeys your commands. In combat, it rolls its own initiative and acts on its own turn. A familiar can't attack, but it can take other actions as normal.
What dictates what these actions are? Based off the other thread, it seems that because the familiar is of the spirit, it therefore can understand your intentions perfectly.
If that's the case, I have 3 questions based off this particular set of circumstances:
Does Familiar telepathy work when unconscious? If so;
Can I direct my familiar to take certain actions while being unconscious? If so;
Can I have my familiar cast a touch spell like Spare the Dying cantrip while unconscious?