I always leave mine open until the fire is completely out. Usually,
from a practical standpoint, this means overnight, as by bed time,
their are usually quite a bit of embers still burning. I don't
understand your comment about the fireplace having another path up the
flue besides the passage with the damper. I've never seen that in a
home fireplace and it would be a big energy waste if it were built that
way.

As for the CO issues, people have died in tents using charcoal grills.
Given the air volumes, it would be harder to do this in a house, but
why take a chance? Plus, fires that looked like they were out have
been know to spring back to life hours later.