“Can/Should I Still Train When I’m Injured?”

It totally depends.
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If your injury is severe, you definitely should receive appropriate treatment and follow your doctor’s advice/recommendations rather than some random BJJ bro’s advice on the internet.
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And even if your injury is no that serious, it’s still a good idea to give some rest to the affected area of your body.
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I bet, for example, you will get your finger/s jammed from time to time during the course of your BJJ career. Or you might tap a second too late and hurt your elbow, but it doesn’t feel too bad.
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Again, the most reasonable thing is to give that area some proper rest, but in these cases, you kinda feel silly to skip classes altogether just because of a finger that got jammed a little bit.
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In such cases, I would still train based on the thought that my minor injury can allow me to work on positions/moves that don’t require me to use the injured part.
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For example, a while ago, I hurt my fingers… probably because I played the spider guard way too much without taking care of my fingers enough. I wasn’t able to make a proper grip.
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That’s when I started working on the single X guard because I don’t need to rely on my fingers so much when I use this guard. In short, my injury gave me an opportunity to work on something I hadn’t considered before.
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Whether you rest completely or keep training is up to you (provided your injury isn’t that serious, of course), it’s good to know that you could still be creative about how you train when you have limitations like minor injuries.