You need to talk with the instructors of the schools near you to get that answer.
If you take losing bad that is something you need to work on for yourself.
16 is not too old to begin karate or any other martial arts. Many start in their 50's or older. Even though you see young children in martial arts they were not created with children in mind. They were meant for adults.
It is great that you are in decent shape You don't need t be to begin martial arts training. The reason it is good is because it is healthy for you. Even though you are in good shape now it is likely you will get in better shape as you train. The conditioning for karate is different than gymnastics. The conditioning for judo is different than boxing. The conditioning for boxing is different than karate. This can go on and on.
"I have been told by some people that if you take karate you MUST compete."
Were they karate practitioners? No, it's not mandatory across the board that you have to compete. The thing is, karate is not homogenous. There are about a dozen major systems, some of which don't even resemble each other, as well as dozens of lesser-known systems and sub-systems. Past that, how a school is run is basically up to the instructor. Some may conform to an organization's standards, while others might not be part of ANY organization.
"Also at most places do they have weapon training? If so what type of weapons do you mess with?"
Please don't refer to weapons training as "mess with". Again, it varies. Some places don't teach weapons at all. Those with a strong connection to Okinawan martial traditions may teach Okinawan Kobudo, which includes weapons like the bo (staff), nunchaku (flail), and tonfa (impact weapon resembling a police night stick). Others may teach Japanese swordsmanship. Others might even incorporate weapons from other cultures, like the stick and knife work of the Filipino martial arts.
"Before signing up for anything do I have to have some muscles built up?"
Being in shape helps, but being in shape for martial arts is something you have to train in martial arts to do. If you're already doing some sort of physical conditioning, keep at it; if you're not, start, maybe by running, "boot camp" style workouts, and light weights.
"By the way I am a 16 year old girl, and one of the many reasons that I had not taken karate already is because I thought that I was too old to pick up on something like Karate."
I don't know why people have this notion that if you start martial arts after age five or something, you'll never learn it. My father was about four decades older than you when he started out. There were things, due to his age, that he couldn't do as well as younger students, but he did "pick it up" well enough to get a black belt after a few years.
Some schools and instructors force their students to compete and others don't. I myself never forced students to compete nor were their promotions ever based on that. The only requirement that was levied on them was that they had to go and watch a tournament before they could be promoted to brown belt or 3rd kyu. That way they would at least have a small amount of first hand knowledge about this at least but otherwise it was never mandatory for any of them to actually compete.
Instructors who try to force their students to compete usually are either trying to pump up the numbers and reputation for their school and increase business or just trying to inflate their own ego through the hard work and sacrifice of their students. A good instructor will never demand of a student something that they themselves have not done before or are currently not willing to do. However there are those out there, and parents as well, who force their children into things that they themselves would not do or have never done just to inflate their own ego and sense of self worth. I have had that conversation with more than one parent or instructor before. Forcing students to do something that maybe they can't really afford to do or don't want to do like what you ask is manipulation and abuse in my opinion. While I view competition as an important facet in some student's development and growth it is only one aspect and should never be forced on someone.
As for your age you are by no means too old to take up some martial art. Actually you are at the ideal age in some ways because you are probably still flexible and physically active and old enough and mature enough to take instruction pretty well. People tend to lose their flexibility and become less active when they get out of high school and get older and also maybe don't have the time for it then either. Shop around and look for a good instructor that has high standards and that offers classes and training in those things that you are looking for.
There are many different styles of karate.
So you won't be able to get a valid answer.
With that said, the short answer is that in 12 years I was never forced to compete. In fact, other than time on the floor sparring I've only competed when I felt like it.
Like you I don't like to compete, but as I've gotten older and more experienced I actually am to the point where I'm thinking about it as it's something to do and it gives one a chance to practice with a goal in mind. Of course my training as a black belt is very different than yours will be as a beginner. So I suggest you look around, and find a school that you think you'll like. If you do, you'll fall in love with martial arts and make it part of your life.
In point karate, nobody gets hurt, usually. If you take things too seriously, start sparring informally.
Weight training is not specifically designed to help with full body martial movements. It can stunt your muscle refinement, actually. So try not to do too much of that before or after.
Martial arts can be taken for all ages. It's not a kid sport like protege music lessons.
Each school is different: not all compete. So it's not always the style, sometimes, it's the school. You'll need to ask the instructor about what is expected of each student.
So for a while now I have been interested in martial arts. But one of the things that is keeping me from taking karate is the competitions. I have been told by some people that if you take karate you MUST compete. I am mostly just doing karate for self-defense purposes, and at some places they have classes that teach just self-defense. If I go just for the self-defense do I have to compete? I do not like competing in anything because I take things way too seriously, and if I lose I will be mad at myself for months. But after doing the self-defense stuff I might be interested in trying to become a black belt. If I do go for the black belt will I have to compete? Competing I have heard from people is really fun, But I don't like hurting people. Also at most places do they have weapon training? If so what type of weapons do you mess with? Before signing up for anything do I have to have some muscles built up? I know that is probably a silly question, but I just don't want to be the weak one in the bunch. I used to take Gymnastics so I already have some muscles, and I work out at least twice a week. But will it make everything easier if I go in there buffed out? By the way I am a 16 year old girl, and one of the many reasons that I had not taken karate already is because I thought that I was too old to pick up on something like Karate. So yeah, if anybody could help me out with my questions and concerns I would really appreciate it. ^_^
