1) I think other people have covered the passport question.. but I'd certainly get on that first. Processing can take up to 6 weeks (yipes!). You can expedite it for more $$, obviously. The travel.state.gov website should have all the info you
Visas - Americans fall under the visa wavier program, so you don't have to apply for anything beforehand. You'll get a 90-day temporary visitor sticker when you land.
2). It's probably doable. Most touristy areas will have backpackers hostels or minshuku that you can stay in for fairly cheap. I typically pay between 2,500 yen - 4,500 yen a night, depending on where I stay and what type of room I get. I usually budget 7,000-10,000yen per day when I'm traveling for food/souvenirs/local transportation (bus/taxi, etc)/admission fees and that is usually plenty.
3) Generally in-country exchange rates are better than the rates you'll find at airports in the States. Whenever I go back and forth I usually just change a couple hundred dollars at Narita... then use my bank card to get at the rest. All Japan Post ATMs take foreign cards (no problems), and the ATMs in 7-11 take them now too...
Japan is a cash based society... while it's slowly changing, it's a lot less hassle than having to deal with your credit card, obviously. You'll be able to use your credit card at hotels and large stores (like big department stores and the electronics store, etc). But for the grocery store, convenience store, etc. expect to pay in cash. Don't worry too much about carrying large amounts of cash with you... everyone does it, and you're pretty safe doing so. And ditto to the thing about calling your card companies.... heck, I've had one company cancel my card for foreign charges even though I had called and told them (several times) that I live overseas (grumble grumble).
4). All 4 main islands are connected by rail... and the rail pass that you'll want is probably this one:
http://www.japanrailpass.net/ which is good nation-wide and valid on the shinkansen. There are regional passes available as well, if that would fit your scheduling better... but the national pass will save you a bundle if you're planning on going to lots of different places.
There's also the Seishun 18 Ticket, which is good nation-wide but on local trains only (no Limited Express or Shinkansen)... it's 11,500 yen, but you can only use it during certain periods, so it would depend on when you're going to come.
Highway buses are another option, and while they take some time, they're fairly cheap. I went to visit a friend in Okayama prefecture last spring and paid around 17,000 round trip for the night bus. This is cheaper than one way on the shinkansen from Tokyo to Okayama. Buses from Tokyo to Kyoto/Osaka start around 4,500 one way.
Flying domestically is kind of expensive. Tickets to Hokkaido start at about 10,000 yen one way. If you're going to be here during a peak travel season, expect to pay a little more.... probably closer to 15,000 minimum, one way.
5). I spent 6 days traveling around Kyushu over spring break and it was probably one of the best trips I've had in Japan... definitely a favorite. Very different atmosphere than where I live (Yamanashi) and the people are incredibly friendly. I have yet to make it to Northern Japan (that's on the list for this summer), but besides the obvious Tokyo/Kyoto/Osaka.... my favorite places that I've been are:
Nikko (couple hours north of Tokyo... doable in a day trip if you start REALLY early... but two days is probably best) Beautiful scenery, great temples. You can get a money saving train/bus pass from Asakusa.
Fujigoko Area - I'm a bit biased towards this one, since it's about a 40 minute drive from where I live, but there's lots to see, some really nice onsens, scrumptious hotou, and of course, Mt. Fuji. The train to get there is expensive, but there's several buses that run from Shinjuku.
Mt. Aso - Active volcano in Kumamoto Prefecture on Kyushu. You can take a ropeway right up to the crater. The surrounding area is pretty inaka.... the train line to get there isn't even electrified, which was a definite change from what I'm used to.
Himeji - Home to one of the most impressive Japanese castles... doable in a day trip from Kyoto/Osaka
Matsumoto Castle - Another nice castle in Matsumoto, Nagano Prefecture.
Kamakura - Old capital of Japan, south of Tokyo. Lots of great temples.
I use japan-guide.com to get ideas for where I want to... then typically turn to my Lonely Planet/coworkers for more ideas. I'd be glad to help you with ideas, just send me an email. I'm most familiar with Tokyo, Yamanashi and the surrounding areas.