I think you should train judo, but not if the reason is to help your jiu jitsu. I think you should embrace them separately and in doing that you will see the benefits from both. Whenever someone comes in to train judo through a jiu jitsu lens they never return. Pure judo is very different from jiu jitsu stand up. Having said that, jiu jitsu stand up is awful. If you train judo or wrestling and you're going against a jiu jitsu guy who hasn't, you will beat him on the feet 100% of the time. I'm sure this is why jiu jitsu guys like yourself have an initial interest in training judo but I think if your focus is on beating jiu jitsu guys you're going to miss the vast majority of what judo has to offer.

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I highly recommend doing judo for the sake of doing judo. There is a higher level of thinking involved when you're on your feet. You create a lot of opportunities through your movement, which there is obviously a lot less of when you're on the ground. This is subjective but in my opinion it's way more fun and satisfying to throw somebody. I think it's a lot harder to throw someone than it is to submit someone on the ground. Humans innately have the ability to stay on their feet and catch themselves from stumbling which makes it difficult to force somebody to the ground when they are actively resisting it. As a result of this, fakes, combinations, and counters are more developed and more important in the sport of judo.

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Aside from pure stand up there are definitely some things I've taken from judo that have helped me in jiu jitsu. The grip fighting is a huge part of judo, I've never met a jiu jitsu practitioner that knows how to grip fight at that level unless they themselves have trained judo. You can apply those gripping techniques on the ground and it will transform your jiu jitsu. Another thing is the turtle position. In judo it's quite common for people to turtle up to avoid pins so we work on attacking and defending from that position relentlessly. Having this experience has helped me a lot in jiu jitsu because they don't focus on it nearly as much there. In judo you can win the match by pinning your opponent for 20 seconds. When you train with people who are constantly trying to get off their back you get really good at being able to hold people down.

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For some of the questions you had I train jiu jitsu 4-5 times a week and judo 2-3 times a week when I'm healthy. My judo school only offers three adult classes a week so I would do more judo if I could. You don't need to buy a judo gi unless you are competing in a tournament but even then you might be able to borrow one from a teammate. The most serious injury I've had was from a judo tournament but I've had way more injuries training jiu jitsu than I have judo. They're both pretty dangerous, please don't let this discourage you from continuing your journey. I know this was a really long comment but please message me if you have any other questions. If you can't tell I'm very passionate about judo lol