Anyone with an International Driving Permit (IDP) issued abroad under the 1949 Geneva Convention on Road Traffic can drive a car in Japan in accordance with Japanese laws.
Only international driving permits issued by a signatory to the 1949 Geneva Convention in conformity with the Convention’s provisions are recognized as valid for driving in Japan. An international driving permit on the other conventions or treaties, including the 1969 Vienna Convention, is invalid even if it is issued by a signatory to the 1949 Geneva Convention.

Please refer to the list of the contracting states of the 1949 Geneva Convention.
List of the contracting states of the Convention at Geneva 1949

It depends how long you stay abroad/outside Japan. In case you reenter Japan in less than three months after leaving the country, you shall not drive with your international driving permit even if it is a new one you have obtained during such a stay outside Japan. On the other hand, in case you are a temporary visitor or a resident in Japan, who is not registered on the Basic Resident Register (Jumin Kihon Daicho), you may drive with your international permit regardless of the period of your absence in Japan prior to the entry.

No. You need to have a stamp/check for the Category A so that you can drive a moped or motorcycle less than 50 cc.

No. That shall constitute a violation of the Road Traffic Act of Japan; driving without a license.

You can drive if you are a holder of the valid license issued by the competent authorities of Belgium, Estonia, France, Germany, Monaco, Slovenia, Switzerland, or Taiwan. However, you need to carry, along with the license, its Japanese translation prepared by the embassy of the issuing country (or the Taiwan-Japan Relations Association in case of Taiwan) or the Japan Automobile Federation (JAF).
Valid Foreign Driving License in Japan

Yes. Depending on the issuing authorities, however, the place to go for such procedures may differ.
Click here to read instructions for converting your domestic license to a Japanese one