I'd advise against a carousel. I don't have sources, but I've read that something like 90% of viewers never make it past the first slide of a carousel. They just aren't good UI. ++ for all your other links though.
I started programming around a time similar to you, and I also faced many of your challenges. I know this may not be the general opinion, but my opinion is to not worry about starting a web design business. This is a super competitive field, and your age WILL hurt you when it comes to finding clients.
That being said, I started working for a web development firm at age 15. I didn't have a portfolio and I didn't have an impressive interview; I simply knew the right people and was in the right place. I lucked into a great job. I was able to work after school and on weekends, and I'm still employed at this firm. Working for a firm allowed me to mask my age. I wasn't a person; I was a member of a team. I produced the results expected of that team, and clients never once asked my age. I'd continue networking with people and being active in the community. Eventually, you will find the right opportunity -- sometimes it just takes time. Connections will be your biggest asset as this age. See if you have any family friends who own businesses who might need a website.
I never viewed web development as a career, but rather as a job. My passion is computer science, and by working in web development for several years before going to college, I've had the opportunity to gain communication skills and work skills. I am currently studying computer science, and I love what I'm doing. I feel if I was focusing on both computer science and web design, I would be spreading myself too thin. I'd suggest picking a field of focus, and going full throttle.
I'd suggest spending more time learning and less time trying to start a business, especially if you're facing resistance with the business. Running a business is a huge time investment (if you do it right), and this time might better be spent learning at your age. You have the blessing of being at an age with minimal obligations. You should use your free time to learn everything you can. Watch TED talks, read Wikipedia, take online courses, etc. This will broaden your horizons and perhaps open doors for you.
Finally, I'd suggest changing your personal website -- both design and content. I'm not accusing you of copying (and I'm not sure who did it first), but your website looks like a direct copy of Macaw's sneak peek website -- http://macaw.co/peek/sample. While a fraction of the design community might be the only people who recognize this, it still might give people the wrong idea. Considering you want to be a web designer and a website nearly identical to this was made in a matter of minutes as a demo, you might want to show something unique and reflective of your skills (your case study and your personal site do not look like they came from the same person). For your case study, I'd suggest explaining why having a site helped this client. Did he go from 0 traffic to 100 unique daily views? Did he go from 100 unique daily views to 1,000 unique daily views? Businesses and clients want to see results.
If you have any questions on web design/development at a young age, feel free to respond to this comment and I'll do my best to answer.