Cat's tail caught in door: Is it broken? When to go to the vet?

Unfortunately, it's not possible to tell is his tail is broken from these pictures. I would have to examine the tail in-person in order to tell this. The wound does look quite deep though and it's definitely possible that it was broken.

These injuries tend to be vary painful, so I would consider going to the ER vet if he seems very guarded and not interested in food - these are signs of severe pain. Otherwise, as long as it's not bleeding profusely, this injury can wait until your regular vet is open tomorrow.

For now, you should flush the wound using a syringe or other dispensing device with warm water for five minutes. This may not be easy, because cats tend to hate water getting sprayed on them. I recommend having someone help you with this and wrapping Alexander in a towel to make sure the wound gets flushed well. Below is a link to a demonstration of an eye flush that shows how to restrain her well.

https://www.youtube.com/watch?v=tDGzdUTUdNI

Infection is the biggest concern with these wounds; so after flushing, it's best to apply an antibacterial cream to the wound. In order to keep the wound clean, it's best to flush the wound 1-2 times a day and gently wipe away any big crusts - make sure not to scrub the wound, because this can make it worse. Apply the antibacterial cream after each flushing.

I hope that helps and Alexander gets better soon.