Oh my gosh, YES! If you're really interested in graphic design today there's no need to wait for formal training at a trade school or college. Play around with design right now. Pay attention to any visual or functional design all around you. I include functional design because often the two are related in the final product. Aside from paying attention to interesting designs around you, you can give yourself "assignments" right now. I used to look through magazines and websites that had ads that I thought could be improved. So, that's what I did. I redesigned the ad myself. It's excellent practice. Gives you a quick starting place. And shows you just how hard it can be to produce visually appealing designs that also convey the desired message.

Hang on to your favorite "assignments" for two reasons. One, it shows that you've been interested in Graphic Design for quite awhile. And two, it's great to look back at your old art and see how far you've progressed! I do recommend getting additional training after High School. Do some research, though. You don't need to spend a ton of money (and end up with a huge debt). A well-equipped technical program that trains you in the latest software is important, as are the teachers and their work experience. But sometimes you can find amazing programs at local community colleges. WAY cheaper than private art institutions! In the end, after learning the software and design skills, it's YOUR talent that will advance you the most. Not necessarily the pedigree printed on your resume.

p.s. (once you've learned a thing or two, consider testing your skills with a small start-up company that needs some graphic design help. Doing a few projects at "student rates" for them will help the company with their expenses, but also show that you've gone beyond school assignments and have actually done some professional work!)

You should definitely have side projects that you are passionate about. Doing client work can be fun, but might not fulfill specific passions you have. Practice as much as you can in your spare time, and if possible get a design internship to expand your portfolio.

Yes, yes and yes.
What employers want to see is not only student work, but personal work. They want to see if you do more than what is expected of you. And know that you are passionate about it.

Absolutely you should do work outside of class assignments! I would also suggest keeping a sketch book, and doing projects in multiple mediums. A diverse and well rounded portfolio will always be more attractive to prospective employers than one that is entirely focused on one single form of artwork.