Our son shut the cats tail in the door. He had a crook in the tip of his tail the rest of his life.
I would think the cat would keep the tail clean and should be ok. We had a cat shot by a bb gun, he had a small wound on his shoulder. The vet had us put antibiotic gel on it twice a day until it started to heal. Your cats wound doesn't sound that severe though so I would just keep an eye on it for infection.
Oh and don't feel bad, sometimes kitties are very fast and sometimes not so much. I bet he won't linger in the door anymore!
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