Dia dhuit!
> 1
I reckon it will. The timing might need thinking for when it starts and ends. But you'll certainly get from Tokyo to Hiroshima by shikansen with it.
I haven't done it but my lonely planet guide says you can get ferries from Shimonseki and Fukuoka also on the shikansen. About 13000 one way. Mind you its 14 hours by ferry or 3 hours by high speed hydrofoil, to Pusan.
Off the beaten track or tourist trail might mean trying some side trips into the countryside a bit more. Tokyo, Kyoto, Hiroshima are popular destinations. partly because they are easy to get to.
> 2.
I found travelling between places tended to soak up a day. sometimes because of the whole check-in to Hotel after 2 PM thing. I find I don't like travelling but prefer being somewhere.
2 things might help here. Coin lockers and left luggage. These work in Japan. They just don't exist in UK or Ireland anymore. If booking hotels / hostels in advance chose a later check in time. (I'd also say stay close to station area for convenience) Ditch your luggage at the train station or arrange to leave luggage at hotel before checking in. (If leaving and going back to the same hotel you can ask them to hold on to your main luggage for you while you head off with a lighter overnight bag.) Sightsee. Head back around 5 or 6 when tourist sights shut down, check in to hotel/hostel, relax in public bath for a bit head out for any nightlife. Shops open at night ('till 9 or so I think). Tourist attractions don't. Save shopping for the night.
(Other time wasters can be dithering over where to eat. Just go for it-- don't worry too much about inability to read the menu. ) Maybe you know all this from travel in China.
Kyoto.
Buy the 1 day or 2 day bus pass. The underground doesn't tend to go anywhere of tourist interest except maybe the main shopping streets. Most of the sites are around the sides of the city rather than the centre. (unlike Europe).
Check out Google Earth before you go for a better sense of geography than a map.
I recommend Fushimi Inari Taisya. South East of the City, buses seem infrequent but it may be the time to get the train (local company don't think it's JR) If you climb up to the top it'll take half a day I reckon. Otherwise the must sees are around the Mizudera area. Get up early and arrive before the school groups etc. I had 2 very different views of Mizudera; once early morning which I really enjoyed. Then I went back the next year to a particular shrine in the afternoon and it was terrible! jam packed.
If you go Geisha hunting at night in Gion, you'll have a better chance in the back streets than outside the expensive restaurant with the (other) tourists. The Yasaka Shrine nearby is interesting if a bit eerie at night.
If you like temples etc. there is a lot in Kyoto and nearby. you could easily spend weeks there.
Ohara north of Kyoto is out of most westerners way. I liked it. And stayed in a nice minshuku with a great rotenburo.
If you like cycling it's worth checking out bicycle rentals in Kyoto.
> Going to Hiroshima
If you have time stop off in Himeji for half a day to see the Castle. If you're going to see a castle it's maybe the best place to see one. If you're spending longer or like gardens the gardens next to the Castle are worth a look.
>Nara
I really like Nara. I overnighted here but to be honest it could be done as a daytrip from Kyoto (30 mins or so on the train I think?) which makes accommodation hassles easier. Set out early, prepare to walk a lot.
>Osaka if you're tempted
If you're going to experience city life in Tokyo I'd say give it a miss. SpaWorld is fun though.
3. Sumo
I saw an excursion thing on JTB site yesterday (for Nagoya). They might be able to arrange tickets only.
http://www.jtbgmt.com/sunrisetour/
4. Heat and humidity.
Rain might be an issue in June.
Umbrella is the way to go. Raincoats just make you wet from the inside out I found. If you get wet when using an umbrella generally you dry out fairly quickly after the rain or when you go inside.
I'm not sure about June.
(I went in September, hotter I think and fairly humid, and found I went through a change of clothes each day because of humidity.)
Follow the local custom of schoolkids and carry a small towel to mop yourself off every so often.
5. Hashi -- chopsticks
I don't know about rudeness level but I got a very nice set in a box that are meant for packed lunches. To be honest I never noticed anyone bringing their own chopsticks but I'd say they might just put it down to gaijin weirdness. For some unknown reason they're impressed we can use chopsticks.
Have fun!!