I got my middle crush by a. Door 2 days ago, and their. Is a smart. Pain shooting down my finger. Is it broken?

Very possible. It is common for a door injury to cause a fracture and possible nailed deformity of not properly treated, splinted, or fracture reduced and fixed. At the very minimum this will require an xray.
Possible fracture. If there is bruising and swelling, there could be a fracture. Often when the fingertip is crushed, the tip of the bone may break, and the nailbed may tear. If there is bruising underneath your nail, you could have a nailbed injury. If there is no significant bruising or swelling, it should be safe to observe the finger for a few more days. If pain persists see a hand surgeon for evaluation.