Is it permissible for me to pray Dhuhr at its prescribed time with the intention of praying Dhuhr and with the intention of making up for a missed Dhuhr prayer from past years, knowing that I have twenty years' worth of missed prayers to make up and I wish to maintain the current prayers and their Sunnahs?

Making up intentionally missed prayers is a matter of scholarly disagreement, but the safer approach is to make them up. If this is established, then you are not permitted to combine the intention of performing a current prayer with making up a missed prayer, because each of them is an act of worship intended for itself. You must make up what does not harm your body or livelihood, and busying oneself with making up prayers is more emphatic and takes precedence over performing the regular supererogatory prayers (Sunan Ratibah).