Can a DND 5e “find familiar” fey creature put asleep with a “sleep” spell?

A fey creature conjured by a "Find Familiar" spell relates somehow to the racial trait "Fey Ancestry" of elves, which says: You have advantage on saving throws against being charmed, and magic can’t put you to sleep.

Does this hints, that a) all fey creatures can’t put to sleep by magic, and b) what effect has the animal form on the fey creature, because a ‘normal’ beast can put to sleep by magic—does the animal form makes the fey creature vulnerable to the "Sleep" spell, because the racial trait will not pass into the beast body?

confusing: A sprite as a tiny fey has no trait, that magic can’t put it to sleep.

Additional question: Would a flying familiar like hawk, raven, bat, or owl wake up or die (return to its dimension) when it hits the ground, if it were put asleep in flight?