1.
8 days is a pretty short time, and unfortunately you won't be able to see too much of the country. But, you should be able to make at least a few stops. I think on any trip to Japan, Kyoto should not be missed. As mentioned above, it's the cultural (historically speaking) capital of Japan and has more sights of interest such as temples, shrines, castles, gardens, geisha's, old districts, and parks than you can ever see in even a two week visit. Give yourself 3 days there if you can. From there, you can also take a day trip to Himeji, home of probably japan's most famous and brilliant castle. With such a short time, I wouldn't bother with the usual day trips to Osaka, Kobe, and Nara. You'll see enough urban and cultural stuff in Tokyo and Kyoto.
Climbing Fuji is definitely recommended, otherwise I'm not sure what else you do there. That time of year, views of Fuji are pretty scarce because it's covered in cloud. I was in the region for a week or so last summer and literally saw the mountain for about 10 minutes on the morning I was leaving. But, if you can fit a climb in, maybe as a very long day trip from Tokyo or a stop between Tokyo and Kyoto, it's well worth the exhausting hike. Hakone is also a popular tourist destination for it's natural beauty, outdoor activities, views of Fuji, and of course, the all important onsens. You may be able to fit a stop there in too between Tokyo and Kyoto. Unfortunately I can't share anymore info on this region as I was able to stay with friends, had my own wheels, and almost unlimited time, so I didn't pay attention to the finer details of traveling.
As for finding the "real" japan - you'll see it the second you step off the plane. The nightclubs, shopping malls, restaurants, and glitz of Tokyo, or the kimono clad women of Kyoto, will all be as uniquely Japanese as the 800 year old Shinto Shrine across the rice field from my apartment in the mountains.
2. As for trains, again, since I drove myself everywhere I needed to go, I can offer you much advice here.
3. I highly recommend Ks house Backpackers Kyoto. I've probably stayed at 100 hostels over the last 10 years in various parts of the world and I will confidently rate this as the very best hostel that I have ever stayed at. You will easily meet other backpackers there. They also have hostels in Fuji and Tokyo, which I can't personally vouch for, but reviews are positive, and I suspect they are just as well run.