I believe the word is
(credit to @hexomino , @keelhaul , @rhsquared , and @lee-leon ; some of the below were either directly from your answers/comments, or derived from them)
Earth has one?
Moon (note: there are other possibilities, but this is the only one that fits with the final word)
Half man, half bull?
Amongst Mercury's eight neighbours but slower?
Child of Red and Green?
Yellow (note: you get Brown when you mix these colors IRL, but Yellow when you combine the color codes: (255,0,0) + (0,255,0) = (255,255,0). Yellow is also "in between" green and red at stoplights)
Slimy, slippery but can shock you?
He was raised by apes?
It mimics words but can't comprehend them?
This is at the core of this and all of us?
Soul (note: metempsychosis means "transmigration of the soul", and the soul is also at the core of us humans)
Woolly Dolly was one?
Runs through wires?
Shadowfax showed us it's meaning?
I am air-borne but at night?
Indian has two?
I (the letter)
Every problem has one?
Taking the first letters gives you
which unscrambles to the answer.