An eye?

Cut me for nothing to be missed

Got nothing for this one A slightly dated term - to cut one's eyes at something is to briefly look at it - thanks, @Cubicon!

open me to be surprised

How many surprises start with being led somewhere with your eyes closed?

Be my fruit and you'll be loved

She was the apple of her father's eye

Though if I'm from an animal you'll be doomed

Bullseye, if someone's aiming for you

An eye?

Cut me for nothing to be missed

A slightly dated term - to cut one's eyes at something is to briefly look at it - thanks, @Cubicon!

open me to be surprised

How many surprises start with being led somewhere with your eyes closed?

Be my fruit and you'll be loved

She was the apple of her father's eye

Though if I'm from an animal you'll be doomed

Bullseye, if someone's aiming for you

An eye?

Cut me for nothing to be missed

Got nothing for this one

open me to be surprised

How many surprises start with being led somewhere with your eyes closed?

Be my fruit and you'll be loved

She was the apple of her father's eye

Though if I'm from an animal you'll be doomed

Bullseye, if someone's aiming for you

An eye?

Cut me for nothing to be missed

Got nothing for this one A slightly dated term - to cut one's eyes at something is to briefly look at it - thanks, @Cubicon!

open me to be surprised

How many surprises start with being led somewhere with your eyes closed?

Be my fruit and you'll be loved

She was the apple of her father's eye

Though if I'm from an animal you'll be doomed

Bullseye, if someone's aiming for you