I don't think there's anything wrong with you. In fact, it sounds like you're already pretty good at dealing with rejection, which is something that a lot of actors and writers struggle with.
I'm sorry that you were made fun of in elementary school for your singing and that it's affected your confidence. That kind of bullying is commonplace for schools and anywhere someone grows up, but it still sucks. Unfortunately, when everyone around you is a kid, you can't really depend on them all to behave like adults. I strongly suspect that the people who made fun of you just wanted to make fun of someone, that you were noticeable because of the talent shows, and that you were a singer was all they knew about you, so they made fun of your singing. It seems likely (based on the little I know from your question) that the abuse was less personal than you think it was, even though it was so painful. If it's any consolation, I doubt they knew anything at all about quality singing. :-)
My advice to you is to continue to study and practice for as long as it interests you. Your confidence will improve with experience, and your focus is likely to broaden, narrow, or redefine as you're exposed to more of these arts.
Take classes in acting, singing, and writing. Your instructor will be able to evaluate your abilities and guide you toward improvement. Other students might have feedback, but probably aren't very experienced either, so take or leave whatever they say depending on how much you trust them. Take assignments seriously and pay attention to the lessons and to the work of other students. Ask questions if you have them. Make appointments for more specific feedback if you feel you need it. Don't worry about what anyone thinks--you're there to learn, and you want to get better because you care about this stuff, dang it.
Get involved in projects. Audition for shows. Sign up for concerts. Enter writing contests. And do your best with all of them, no matter how small your role. And don't be afraid to try out for things you haven't done before. That's how you learn what you want to pursue.
Also, maybe even first, be an active audience member. Go see plays. Watch critically-acclaimed movies. Go to concerts. Read a lot, particularly in the genre or format that you want to write. You'll learn a lot, you'll see what works and what doesn't (and you should think about why they do or don't), and it will (hopefully) keep you excited about participating and doing the best you can in each of those areas.
As for worrying about others and their opinions or judgments, all I can really say is to try not to. If you consistently do good work and conduct yourself professionally, you'll earn respect as a solid and reliable performer from the people whose respect you want. The people who won't respect someone just for that lack the full capacity to respect others, anyway, and you'll eventually get away from them. People are likely to say both good and bad things about your work that you may or may not believe, but you're the one who will know whether or not you tried your best.
You're 16. That's really young, even though it doesn't feel like it sometimes. You're allowed to try out different things and study what you're interested in. There's a lot of other stuff that young people have to study, too, of course, and you should take care to not let it slide. Getting good grades in all subjects will expand your interests and abilities (good actors and writers are smart people), will make getting into a college of your choice easier (where you can study more of what you want), and tends to make a good impression on parents/guardians, whose support you still need.
Good luck!