Unless there is some key piece of information you have not shared, divorcing will not make her eligible for legalization. For a comprehensive case analysis, I would encourage you to schedule a private consultation with an attorney.
It is impossible to answer your questions without knowing the dates when the various events described above happened. Derivative citizenship law is complex and the rules depend on when various events took place. Your best bet is to schedule a consultation with an immigration attorney and bring copies of birth certificates of the people described above.
Provided that your spouse is a US citizen, you should be able to adjust without having to leave the United States. However, there are still timing and preconceived intent issues to be resolved. You should see an immigration attorney for a consultation before taking any further steps.
As the parent of a US citizen, as long as you made a lawful entry, you should be able to adjust your status to that of a lawful permanent resident (get your green card). You should consult with an immigration attorney to determine whether there are any disqualifying factors in your case, such as certain crimes. The process for adjustment of status is currently taking about 6 months.
Overstaying a visa can affect your ability to obtain a visa in the future, however a 60 day overstay should not be grounds for a denial of an immigrant visa petition filed by your US citizen spouse. That said, there could be ramifications if you overstay for a longer period of time, and I would advise that you consult with a US immigration attorney before you leave the country. We cannot advise you on what consequences, if any, your overstay in the US would have under Japanese law. You would...
Lawful permanent resident status is intended for those who wish to reside permanently in the US once their status is granted. If your spouse cannot do that or does not wish to, it would probably be best to delay the filing of your application until she is prepared to make the move to the US on a permanent basis.
You haven't provided enough information here for us to answer your question. If you want to "do it right" your best bet is to hire an immigration attorney to review the individual details of your case, advise you whether it is prudent for you to file an application for immigration benefits, and, if so, represent you in the process. Incorrectly prepared cases can cause significant delays, denials, and the loss of your filing fees.