To answer your questions first:
The easiest way to learn the language is to enroll in a language immersion program such as the summer ones offered at Temple University (http://www.tuj.ac.jp/english_program/index.html). Keep in mind these can be very expensive, but I believe you can use enrollment in such a program to obtain a student visa. Knowing Japanese is not a requirement for living in Japan (I certainly got by), but it makes things a lot easier--especially when looking for work.
How much money should you bring? That depends on where you want to live. If you want to live in Tokyo, expect to be paying about $800 a month for rest (not including utilities). Keep in mind that there are additional expenses to renting in Japan including key deposit and sometimes rent paid up-front. I lived in a boarding house where expenses were shared (http://www.conveniencehouse.com/En/). Japan is a very very very expensive country. I lived there for only 3 months, and though I was traveling a lot, I still spent about $2000.00 (not including rent and my commute--which was paid for by my job). I was living on the cheap (i.e. getting bread from the 100 yen store and drinking liters of tea at a time). Traveling within the cities isn't cheap either (as residents we can't get JR passes--even if you have the right visa they don't fit our needs well).
If you live in the countryside or away from major cities your living costs are lower.
What age should you move there? As many people have mentioned, you often need a 4 year college degree in order to secure employment in Japan. The one exception to this rule, I have heard, is teaching English in a language school. In major cities, language schools require a 4 year degree, but my friends tell me the standards are lax in smaller cities and less-densely populated areas. Also, many people have said that to work in Japan you need a secured job and a working visa before you visit--this is not the case. Many of my friends who I lived with entered Japan with a regular Temporary Visitor visa, found jobs at language schools/nurseries and sometimes as nanny's, and were able to obtain working visas with the help of their employers. My advice: stay in the U.S. and complete a 4 year degree, then find a job with a language school in Japan in an area that interests you (you may want to visit before hand to see where you would like to live). Many of my friends who taught English had plenty of spare time--which you can use to take language lessons, art classes and begin your career as a graphic artist.
Yes, anime/manga/videogames are a large part of the culture in Japan. Keep in mind though--in Japan work is taken very seriously. Don't let the popularity of anime/manga/videogames confuse you into thinking that living in Japan will be all fun and games. If you want to make it in Japan you need to work very hard. I highly recommend visiting the country before you decide to move there (you could even study abroad there while in college in the U.S.!)
Where should you live? I think I mentioned earlier that it will depend on your finances, the availability of jobs, your skills (including the possession of a college degree) and your own personal desires--Japan offers the countryside, large cities and small cities.
So like I said: you're probably a junior in H.S., right? Study hard now, go to a 4 year college (or a community college and transfer), visit Japan, maybe take some Japanese classes in college and some studio art classes to hone your skill...then if you still want to come, you'll know what to do.