Hello,
Congratulations on being able to visit Japan and do something you've been wanting to do since you were 8 years old. Definitely a very exciting thing!
I can help answer a couple of your questions.
As far as bringing a sword back with you, I was under the impression it wasn't a problem because many souvenir shops have signs that say 'OK to take home' and 'Ok at airport', etc. However the webpages below beg to differ:
【老品牌】信誉平台
http://www.kendo-world.com/forum/archive/index.php/t-221.html
It appears that it depends on what kind of sword you're buying. There are different kinds, levels of authenticty, etc. Some require paperwork, some don't. I'd probably figure out what kind of sword is most easily transported and then make sure you buy one of those.
As for climbing Mt. Fuji, I have to disagree with the above poster about it being for professionals only. Many people, Japanese and foreign, young and old, make the trip every day. I went with 5 friends in August which is perfect climbing weather. If you're going later in the year it may be impassible due to snow and ice; the above poster may be referring to that. I know that past a certain time of year the guided tours and 'tourist' element dies off and hardcore climbers are left on their own. That's all I know off the top of my head, you'll have to research it more. But if the weather is fine, the trip is definitely worth it, it took my friends and I six hours and the sunrise was WELL worth the trip. Bring food, water and layers of clothing, it's cold at the top and you grow hungry quickly. To get there take the bus from Shinjuku station, it takes about 1.5hrs and puts you at the 5th station where the climb begins. Cost is 2000en round trip I believe. The bus loads on the street near a large camera store, so you'll have to exit the station and look for the ticket office. But its quite a famous attraction so you won't have much trouble finding it.
Regarding your extra sight-seeing days, I always recommend Kobe to those who are near the Osaka area. I personally found Kobe to be a very nice portside area. The earthquake museum is a definite must-see, a lot of hard work and effort went into creating an information center for one of Japan's worst earthquakes ever. You can get there by bus from Sannomiya station, Kobe's main station, which is a 40min train ride from Osaka. The city itself, having been largely rebuilt in the last decade and a half, is quite modern looking (in my humble opinion) but has a nice mix of tradition and tourist interests. I recommend a boat cruise around the harbour area if you like to see big ships up close.
In Osaka, there is also an aquarium and of course Universal Studios Japan if you're into that. That's something I'd recommend doing only if you've got money and time to burn, the aquarium however I highly recommend. I always forget the name of the station, but I believe it's at Osakasa station or something to that effect. Also in Osaka, check out the Namba area. Lots of interesting restaurants and shops there, quite a lively strip with lots to see and do. The eye candy is especially good in the evenings when everything is lit up.
The Miyazaki museum is, if I remember correctly, in the Kichijoji area which is at the end of the Keio/Inokashira line. That line connects Shibuya and Kichijoji, passing through Shimokitazwa/Odakyu line, in Tokyo. If you're spending time in Tokyo I recommend stopping at Shimokitazawa and checking out the area, it may be gone in the next few years due to redevelopment and there is a plethora of interesting shops and restaurants there.
Re: the museum, there's a way to get there from Shinjku too but I always went the Keio route. That's kind of the long way though. You have to walk through a large park to get there. Here's an article I found on it, the routes they recommend are probably better if you're not feeling adventurous with directions:
Japan
As it states, you cannot just go there the day-of, you must buy your tickets in advance from a Lawson's convenience stores. Many event and show ticket purchases are dealt with through convenience store kiosks in Japan. Therefore, unless you're able to secure tickets before arriving, your time frame may not alllow you to reserve tickets if its busy. I know mine and my gf's at the time took a couple of weeks.
Regarding your concerns of 'ethnicity' in Japan, in general I do not think any of the places you've mentioned will give you trouble. In more traditional areas especially, the rural lifestyle and surroundings lend themselves to less foreign visitors and some residents prefer to keep it that way. There are many differeing opinions on this forum as to just how discriminatory or not discriminatory Japan really is towards gaijin, but overall your standard tourist route shouldn't pose any issues. Carry a phrase book, make an honest attempt to be humble and low-profile (as is the Japanese way), and you'll fit in well. Keep in mind that foreigners get a rough ride in any country and try not to let anything spoil your trip if it does happen. That was my key to enjoying myself there.
Tokyo nightlife.. there are lots of clubs in Shibuya but many of them require I.D., and you won't have an alien registration card so you may not get in. I had trouble getting in without mine. (I forgot my wallet one night, I had some ID but it wasn't Japanese). If you want night life I'd head to Roppongi, this is an area with tons of foreigners and so the industry is fairly foreigner-friendly. The flipside of that is that it's not very 'traditional' and there are less Japanese people in the club! But entry will be easier. Drinking in the streets is somewhat acceptible. You won't be hassled by the police but out of courtesy I believe the Japanese prefer that you brownbag anything stronger than a normal can of beer.
Sorry, don't know anything about that mountain you mentioned I have never had the experience. Maybe someone else can chime in there.
Hope that helps, please check out Jref's resources if you haven't already. (top of the page, click 'Practical', 'Society', 'Culture', etc..)