1. March, April and May are fairly pleasant times, as is September and October (although you are risking typhoon season then). Summer is unbearably hot and humid in the city.

2. I would say a week is more than enough in Tokyo if you just want to do touristy things. You can also squeeze in a couple of away days from Tokyo to the national parks or Mt Fuji.

3. Kyoto seems to be the main other place people talk about. I've been advised in the past not to bother with Hiroshima. If you really want to see one of the bomb sites Nagasaki is the more interesting.

4. Can't help with this one.

5. Nor this one.

6. They are both daylight flights so I'm not sure it'll make all that mich difference. If you want to hit the ground running in Tokyo I'd do F outbound where you'd also benefit from western catering being loaded. If you expect to be fairly pooped at the end of your holiday them F home would allow you more rest although the catering can be a bit Japanesey.

One more point. I don't know what the rules are for redemptions on your ticket but it's worth bearing in mind that the Narita service may still be down at a single flight per day in March/April so seats may be a little harder to come by.