What has four letters and starts with coal?

I'm just a branch, with a magical PR team.

A: 'Let's go for a ride! I'm picking a horse.' B: 'Yes, I want a horse too!' A: 'Haha, you're behind me, now you'll never catch me!' B: 'So! Only because the horse next to you was already taken.'

I’m always around on a sunny day. Pan grew annoyed when I ran away. Cast over your face means you’re feeling dismayed. I need another line so hooray for croquet.

Dicks in black and white, often in steamy scenes. Ideally private.