Hi there!
I have never been to Burma myself but with what I have read of it, I can imagine that it is a very big change compared to Japan.
For you first question:
I found a couple of Opera Halls in Tokyo and not one that was called Tokyo Opera House exactly, so could you tell me which one it is? (the ones I found include New National Theatre, TokyoTokyo Opera City and Tokyo Nikikai Opera Theatre). If you don't know which one tell me the name of the concert and composer and I can probably find it with that.
Looking at your wishing, I think you will like Kyoto very much, building are a lot lower than in Tokyo and much more traditional buildings remain.
If I understand it correctly, you will start to travel in Japan from the 20th and be back in Tokyo on the 23rd? Allowing for 4 days of travel. In this case I would go to Kyoto on the 20th, stay there for 2 nights and travel in the morning on the 22th to Hiroshima to spend the day there and take a late noon train back to Tokyo on the 23rd.
For this itinerary a 7-day JR Pass would be very well suited, you would have covered all of your travel by JR train except for one day in Tokyo and make great savings too. If you would like I can make an oversight of normal ticket prices.
An alternative to trains are Night Buses, they are very cheap and can cost as low as 4,000 Yen for one way between Tokyo and Kyoto (and save on accommodation costs too) but are very uncomfortable and if you can't sleep in the bus you may loose the next day to catch up on sleep.
In Kyoto hotels can be quite expansive, alternatively if it suits, you could stay at a Hostel, they are very good in Japan and start at about 2,000 Yen/Night.
Or look for a more traditional stay at a Ryokan. These are traditional inns that sometimes charge per person instead of per room and are generally between hostels and hotels in price range. Agoda is a good resource to start, otherwise www.tripadvisor.com and www.hostelworld.com offer a great selection of accommodations too!
Hope this helps you on your way, Japan is a very easy country to travel in and there are many places available where you can ask for assistance if you require help at any point during your travels.
-Daniel