My finger is throbbing it was slammed in the car door a day ago I have put ice on it and I'm still in pain its throbbing like a heart beat

It might be broken. Ctush injuries to the hand are very common and doors are a huge culprit. If there is persistent throbbing there probably is associated soft tissue and bone injury. I would recommend good x-rays and an exam by a hand surgeon. In the meantime keep it elevated and iced to help with the swelling which is causing the throbbing.
Pain. This is normal reaction after trauma; however broken finger can caue pain to prolong, may need to be evaluate for a fracture.