Assuming you're 15+, you'll be racing mostly Tecnica Cup races if you race as part of NHARA.
As a former/current (at times) USSA/FIS racer and current coach, here's my advice:
Check out the NHARA website at www.nhara.org and poke around and do some reading. Then contact the president of the gunstock club and figure out a time to meet and share your story and get a sense of how you can get involved and if they can help you financially should your parents be unwilling/unable to pay for it. Gunstock has a great club program with racers at every level, from people just starting out to high-level FIS racers. Check it out. http://www.gunstockskiclub.com/page/show/432577-contact-information-for-the-gunstock-ski-club
As to your questions:
Technically you have to be represented at captains meetings, but in practice, roll call is almost never done, and if it is, some coach will represent you, since nobody wants an athlete to miss out because somebody isn't at the meeting. That said, anyone over 18 can represent you in the team captains meeting, they don't technically even need to be affiliated with USSA. They will need a USSA membership to be eligible for coaches' tickets at most races. That said, join a club or befriend a coach who can actually help you here. It will benefit your race career, your skiing in general, and your level of fun. According to USSA rules, every athlete must also be a member of a state organization, so you'll have to join NHARA as well.
As an echo to everyone else, nothing can really replace good coaching and reps on real sets. Since you haven't really been in a USSA program before, and having watched prior video of your skiing that you've posted on epic - you can really really benefit from solid coaching. If you just show up, you might have fun but you'll be way over your head. More importantly, you'll be alone, and one of the best parts of racing is the camaraderie and friendships that form along the way. Join a club. Get a coach.
USSA races are run differently in different regions, but in New Hampshire it's pretty standard. You'll have one run in the morning and one in the afternoon (if you finish the first run). The host club will post a schedule online (for NHARA races, either on nhara.org or on live-timing.com) at least a couple days prior, so you know when to show up, but registration is usually from 7-830 or so.
But really, find a way to join a club or at least get consistent coaching. Not to disparage high school racing too much, but it's not the same. You will really benefit from being in a club.