It is a Winter Olympics event. It is a key that will open many doors. You and your cat or dog each have one.

I’m not a flame, but I set the sky on fire. I’m not a curtain, but I close the day. What am I?

For some I'm an arm, A wooden limb; a chair for a pigeon; For others I'm a military force, A strategic subdivision; I am also choices, Decisions stretching to the unknown; Or a section of a business, That has expanded and grown.

I can run I can me stuffed Don’t pick me up It’s better to blow

My first two is the name for a circle's fundamental ratio, my last two are the symbol for an element, used to make signs stand out at night. Together, I'm a synonym for yearn.