Only the two of us,
Drop to the floor,
Welcome a little girl
Into the door.
What are we?

Mothballs?

Yep!

I received a gift
From a man in a red coat,
But that didn't make me
A hero.
Who am I?

Peter?
"Just because some man in a red coat hands you a sword it doesn't make you a hero! Just drop it!"

Yeah, him. 😉

Cool as a cucumber,
Eyes glowing bright,
I lost my identity
One awful night.

Could it be Shift?
“Cool as a cucumber,” muttered Eustace. “It knows it has nothing to fear.”

Ginger?

Ginger is correct.

*smacks self in forehead* Ginger! Why did I not think of that?

You can take this one, Dilly, seeing how it was your quote that led me to venture a guess. 🙂

Aww...you're so sweet, Ariel! Thank you!

I closed my eyes to the worst event that ever happened
And later closed my heart to all that mattered
Who am I?

... Jadis?

Nope.