I did a 2 months cycling trip through Japan about 1,5 years ago. You might find something useful on my blog, eg planning section:http://bicycletraveljapan.blogspot.com/?m=1
To your questions :
I am pretty sure that renting a bicycle in Aomori and giving it back in Kyushu is not a possibility. You could potentially rent a bicycle eg at a giant shop of Shimanami Kaido or some other destinations (there is one not too far from Narita in Tsuchiura you could check out. BUT their business is to rent you a bicycle for a day or two. Renting long term would likely be very expensive. But feel free to ask them.
Most rental bicycles in Japan are mamachari (if you donft know what that is, Wikipedia is your friend). However searching for ©]Ô^ÂX this page came up: https://aomori-cycling.com/travel-guide/rent-a-cycle-station/
They say that they can answer in foreign languages as well. I take that to mean English.
I canft advise on cycle shops in Aomori as I havenft lived there. There must be some though.
But I would suggest to seriously consider to bring your new bicycles along: you can buy in your home country and get all explanations (I assume you donft speak Japanese, so unless you are a bicycle expert it might be very hard for you to explain the clerk in the shop what you want), you can preorder and are thus not limited to the choice available in that moment in the shop, you can control costs better, do some test rides, get used to your baggage, buy the bags. ...
Frankly I donft see any advantage buying a bicycle locally.
I know people do travel with bicycles in airplanes. (I could say so that I have done so myself repeatedly but a brompton probably doesnft count, but I have friends who did with normal touring bicycles).
If you plan to skip some parts WITHIN Japan by taking a train you MUST have a bicycle bag in which you can fit the entire bicycle. Otherwise it is not allowed. No exceptions for tired foreigners. ( yes, there is one train in Boso peninsula and one by Matsue that seem to allow gnakedh bicycles but this is the great exception.
If you have any other bicycle ride related question I can help with, let me know.
Enjoy your ride through Japan!