My lizard Familiar (from the Find Familiar spell) is often sat on my shoulder. If I use the Blink spell, would it shift with me to the Ethereal Plane, or would it be left behind and splat to the ground back on my original plane?
There’s nothing in the text about things you’re carrying or wearing travelling with you, but presumably they do to some degree. I assume taking another creature would be too much.
With a strict reading of the rules, only "you" "vanish from your current plane of existence" - so you don't take anything with you.
This is discussed in If I am holding an item before I cast Blink, will it move with me through the Ethereal Plane?. It seems clear to sensibly play the game that Blink and spells like it will also transport your clothes, held equipment etc.
If Blink did something as significant as letting you take a whole other creature with you to the Ethereal Plane, it would say so. Unlike letting you take your clothes and held equipment, taking creatures you're touching with you is not a necessary modification to make the spell sensible (and indeed in situations with enemy creatures could be highly detrimental to the spell's utility).
At the end of a fall, a creature takes 1d6 bludgeoning damage for every 10 feet it fell
Assuming your shoulder is less than 10 feet above the ground, your familiar shouldn't take any damage from falling off you as you Blink.
