Train or bus from Hakata station to Nagasaki. If you get off at Urukami, you will be just a little north of the museum, not far at all to walk. If you get off at the Central Nagasaki Station, get on the electric tram and go just a few stops north to the Museum. In either case, the museum will be a couple of blocks plus one block uphill (toward the near mountain) from the stop, as the tram line runs one street over from the Peace Park. The museum is a big red brick cube up on the hill.

No, most 13-year olds are mature enough for this.

Not free, but pretty cheap. Adults 200 yen I think, kids 6-12 half the adult price. It may have gone up, my info is old.

Too far to walk. You can do it by tram but you will need to change. Taxi is easier.

Sorry it's not something a traveller would usually know but feel free to ask the organization themselves

I would suggest 2.5-3 hours total is sufficient for the museum. There can be crowds at some exhibits and all are worth viewing, take your time to read the placards presented in several languages. Arrive early if you're in a rush, and beat the crowds to save time.

Yes, you can get the tickets there.