It took me nearly four years to get a black belt in Taekwondo but it ultimately still felt like I'd achieved nothing. Yeah, I can fight against people my age, but I still felt a little overwhelmed by people larger than me. I want to learn something like Krav Maga but I'm afraid that my age and weight will ostracize me. I have severe social anxiety and depression, and I'm fine with people yelling at me, but I'm not fine with people talking down to me. I'm on the low end of obesity, and I have been trying to eat better but I want to do something more "stimulating," because I have ADHD and exercise makes me want to run into oncoming traffic. How tolerant is the community? Will I do okay? I just don't want to be a burden on anyone.
I think your anxiety and self-esteem are your primary enemies. This is a very real thing for you, I know. I get it. And it's not something you can just snap your fingers and get over. You're a highly sensitive person (HSP) from what it sounds like, and your low self-esteem is probably contributing to your social anxiety, self-consciousness about your weight, and your depression.
You may already be seeing a psychologist for these issues. If not, I highly recommend it. A qualified psychologist can figure out what the underlying issues are and can give you evidence-based treatment that will actually address them head-on. Psychologists don't treat mental health issues with drugs, by the way. That's what psychiatrists do. (Although, your psychologist may refer you to one if he/she sees that it could help.)
Unfortunately, I don't think martial arts alone will be able to solve these kinds of issues. It can help in limited ways. Yes, you'll be stronger, maybe thinner and more muscular. Yes, the right choice in martial art will allow you to be able to actually fight well. But in the end, you'll still have all or most of these issues you mentioned, believe it or not.
So do both.
Okay, that aside, what martial art would allow you to lose weight, gain muscle, and get better at actually being able to fight while keeping your attention from wandering during class and while not bullying you?
Krav Maga is a martial art you mentioned specifically. I'm going to steer you towards something else, because I don't like the idea of young people learning Krav Maga. I explain this at my answer here.
I believe your best bet is to learn Brazilian Jiujitsu or Judo. Both styles are going to teach you how to actually fight, although I think BJJ has the edge due to its emphasis on having a complete self-defense strategy, especially involving submissions, punch/kick defense, take-down defense, and ground fighting.
Right now I believe you're cringing at the thought of doing BJJ. You've seen BJJ schools, and they look really competitive and filled with aggressive guys trying to dominate their opponent. It's performance based, and you're worried you're going to do poorly and make a spectacle of yourself, where everyone will make fun of you and call you names. You're worried that you're not strong enough and not athletic enough. You feel like you need to first get in shape before ever even thinking about joining a BJJ school.
In short, you don't think you're cut out for that kind of a school. You don't think you're the right fit.
You know what? That's your anxiety talking. It's keeping you back. None of that is going to happen. Your first day of BJJ class, you're going to paired either with the instructor or one of the senior students who will take you aside and teach you the ropes. They want you to succeed. They don't care if you can't do a single sit-up, for example. That's okay. They won't call you names or laugh at you. All they expect of you is that you come to class regularly and work at it from whatever starting point you're at. Put in the time and effort, and you'll get better. Each of them were beginners at one point themselves, and they all got better. You will, too. Trust in that.
And one other thing. Resist the temptation to compare yourself with others. You should only look at yourself and your own progress over time. That's what your teachers are looking at, not how you're doing compared with others who started at the same time. They know it's different for everyone.
Some BJJ schools are better than others for this, by the way. Use yelp reviews to help you. Some schools will be more aggressive and more competition based. Others will focus on the fundamentals of self-defense at least until blue belt. I recommend the latter for you. The Gracie Barra schools don't even have you do sport-based anything until blue belt. Until then, they're all about self-defense.
Most BJJ schools have kids and teen classes. At around 16 years old is when they transition students into the adult classes.
Visit some schools near you. And before you start looking, make a commitment to joining a school by some date, say 3 months from now. Write it down on your calendar. When that date arrives, you must make a choice and sign up for your first class by the end of that day.
I mentioned Judo also. Judo has a lot going for it as well. One thing you'll notice is that it tends to have an emphasis on throws instead of ground fighting. That's useful for self-defense, too. And I think most people notice that Judo schools tend to be less aggressive and competitive than BJJ schools are. Judo schools are very kids and family oriented as well. So visit some Judo schools near you, also. See the difference for yourself.
As for physical fitness and losing weight, BJJ and Judo both give you a good workout. But I think the edge goes to BJJ from what I've seen. It depends on the school, but BJJ classes can be quite physically challenging.
How physically challenging it is for you depends on you. You'll hear stories about how new students on their first day of BJJ class had to throw up in a bucket it was so hard for them. Yeah, it's because those students really overdid it. Don't do that. Know your limits. Put in good effort, but don't kill yourself. You'll see everyone doing stuff, and you're going to keep trying hard and failing at first. Don't worry. That's normal. You'll get better over time. You don't have to rise to their level the first day. Or even the first year. You'll get there in your own time.
Hope that helps.
Motivation and Weight Loss
Find an activity that you enjoy. It's unclear what exercising means to you in "exercise makes me want to run into oncoming traffic", but anything you feel that way about is not something you will actually do regularly.
Meditation training, whether sitting, standing, or moving, is part of some martial arts' training. Although it has the potential to profoundly change your mental state, it is also something most people do not have the inclination or patience for. There was a recent study that showed that many people preferred to shock themselves over sitting quietly. When you can stand it, I recommend it.
Being an adult
There is no reason why being young or overweight should inhibit you; it is important that you motivate yourself and show up, have a good attitude, and play well with others. Being an adult is about taking responsibility for yourself and others and not expecting others to do things for you. If you act like an adult, most people will treat you like one.