Is it permissible to make up for the missed fast of the six days of Shawwal from previous years with a single intention along with the Ayyam al-Bid (the White Days), and to pray Duha more than four rak'ahs with the intention of both making up and performing, by analogy with the Prophet's (peace be upon him) making up of missed supererogatory prayers, making up for missed Qiyam al-Layl (night prayer), and the action of Abdullah ibn Amr regarding fasting?

May Allah increase your eagerness for good. As for making up the six days of Shawwal, some scholars hold that they are not exclusive to Shawwal, and that fasting six days in any month is sufficient. Others believe that they are specific to Shawwal and it is not legislated to make them up after Shawwal has passed. The opinion that it is not legislated to make them up after Shawwal is strong because it is a Sunnah whose time has passed. As for making up the Dhuha prayer, it depends on whether it is considered among the regular supererogatory prayers (rawātib), in which case the Shafi'is recommend making it up, or if it is not among the regular supererogatory prayers, in which case most scholars say it should not be made up. Your regret for the good you have missed is a good thing, and rectifying it is by increasing in supererogatory acts of worship.