Moving joints suddenly and with force while having arthritis can cause intense pain and wear damage of the joints over time. A qualified specialist is necessary for evaluating what exercises are appropriate for ones particular condition. And that's what I think should be done in this case. This is a very serious condition that requires expert knowledge of what to do and what not to do. Damage to the joints is cumulative, requiring surgery to replace the joint over time unless care is taken early on.
That being said, the only two martial arts that I can possibly recommend to someone with this condition are: Taiji and Filipino Martial Arts (FMA / Kali / Escrima).
Why Taiji (Tai Chi)? Taiji is a slow moving, meditative martial art that puts emphasis on listening to your body and moving in such a way that it minimizes impact to the joints. It's not a very "mobile" martial art, putting the emphasis on stability first. But is it good for self-defense?
Taiji is a widely misunderstood martial art. Most teachers of Taiji don't even understand it. So finding a good teacher is the first thing you have to do, especially if self-defense is the top consideration. And that may not be easy.
My advice is to look into Chen style Taiji in particular. Find a qualified instructor who can teach not just the form but also self-defense. They are quite rare, so you might not find someone immediately. That doesn't mean you shouldn't even begin. You might try just going with the best instructor you can find locally, and then eventually you can go to workshops, look at videos, and seek out instructors who can take you further.
In Taiji, there are many aspects to it that are useful in self-defense. One is soft skill. That's where you're sensing with touch what your opponent is doing, and you can redirect his motion before it harms you.
Another skill is being stable and unable to be pushed or pulled down. They move in such a way that prevents joints from becoming locked. If you imagine a joint like a hinge on a door, the door can no longer move when the hinge is fully extended in either direction. Whereas, the door can move in or out equally when the door is half-way open. So if you can imagine all your joints being half-way open, you can see that you are optimally able to adapt to pushes and pulls in any direction.
If your knee joint, for example, was completely locked, and someone pushed a little further on it in that direction, you won't be able to move that leg any more. It will be frozen in place. And that means your stability is compromised, your heel will start to come up off the floor, and you'll get pushed over. Taiji has a big emphasis on keeping those joints somewhere in the middle so that pushes and pulls don't cause you to fall.
And there are many other skills Taiji teaches, but the last one I'll mention is fajing. Fajing is roughly translated as "explosive force". With fajing, you emit a powerful push / strike on an opponent, sending him up in the air and back. It must be done at the right time, when your opponent is unable to adapt to it.
Combining just those 3 skills alone, you can defend yourself okay. You won't win fights against street fighters. But you can use it to prevent the most damage to your body and allow you to get away in many common self-defense situations.
Most martial arts aren't going to teach you how to fight, by the way, including Taiji. For that, you need something akin to MMA training. But someone who has arthritis isn't suited for that kind of training.
Here's a video showing a master level Taiji practitioner vs. an expert Judo practitioner. The Judo guy is generally unable to throw or take the Taiji guy down. But this is after decades of the best instruction in Taiji. Most Taiji people never achieve this level. With a good instructor, you can get close to this in about 10 years:
Next, the original poster mentioned weapons training. And the idea here is that if you're physically weak and unable to fight well, then you need a weapon. Weapons are known as "force multipliers" for a reason. A knife, for example, allows a relatively weak individual to overcome multiple aggressive, strong fighters. They give you a big advantage.
Filipino martial arts (FMA) do plenty of weapons training. In fact, you learn weapons before you learn empty-hand fighting in FMA. In Kali, they might give you a pair of escrima first. And that escrima training can be reused in knife and machete fighting. So it's very time efficient and practical. Within about 3-6 months, you will have a good foundation for using a knife or a stick in actual fighting. Stay on for a few years, and you'll have a pretty well-rounded set of skills for self-defense and fighting.
For someone who is disabled or weak in some way, like someone who has arthritis all over their body, I do think it's appropriate to carry a weapon. But here, the original poster is just 15 years old. I don't think a 15 year old has the maturity and experience needed to be able to make good decisions about weapons usage in real life fighting situations. It's not appropriate to use a knife in schoolyard fights against bullies. And doing so will end you up in adult jail with a lifetime of misery to follow, since nobody will hire you after being tagged as a convicted, violent felon. Simply put, weapons and children are a bad mix.
But, I still think FMA training is useful for learning self-defense. And eventually, when the OP is an adult and more mature, I think it might be appropriate to carry a knife, a gun, pepper spray, or a taser for self-defense. Obviously, there are legal ramifications to all of these, and so you have to check first with your local police department before doing anything.
What about FMA training and arthritis? Kali does teach a lot of percussive techniques. You're pounding sticks against each other. You're using punches and kicks. There are joint locking techniques. There are kicks to the knees. All of these have the potential to aggravate existing arthritis conditions.
So you do have to listen to your body. You'll know fairly quickly if you shouldn't be doing something. And you just have to stop, take a break, and maybe even think about never doing that technique or drill again. You'll learn over time what you can do and what you can't. You just have to tell your instructor ahead of time what's going on. They understand. And maybe ultimately you might decide that FMA isn't for you. That's okay, too. But I think based on my own FMA training, your joints will be fine if you're not overdoing it. All of my injuries I got in FMA training were because I was young and wanted to go harder.
Those are my thoughts.
Hope that helps.