Is he sick? (Making sure I don't miss anything)

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Check out this article. Hope it helps.

His tail fin looks like he's been biting it, or it's from fin rot. Since the other fins are fine, biting seems more likely. That can be from stress and/or boredom. Look at videos, like How to Have a Happy Betta, and Betta Enrichment. Do smaller 25% water changes. 50% is more stressful and can cause shock since larger water changes are more likely to cause a sudden change in temperature.