If one is involved in a Mitzva related activity all of Friday afternoon and misses Mincha, how can he "make up" for the missed Shmona Esrei? Is a weekday Shmone Esrai said after the Shabbat Maariv shmona esrei?

"Making Up" a missed Tefilla is (at the very next Tefilla opportunity only) is an option for those who did not deliberately - with no excuses - miss the requisite time framework for that particular Tefilla. I would concur, that total involvement in certain Mitzvot throughout the day, thus causing the missing of a Tefilla, would allow for the opportunity of the "make up" (= "Tashlumin") arrangement provided by the Halacha. "Tashlumin" is fulfilled by the saying of two IDENTICAL "Shmona Esrei" Tefillot, irrespective of the "type" that was accidentally missed. Therefore, upon missing Mincha on Friday afternoon, the "Tashlumin" arrangement is to say two Friday night "Shmona Esreis" - the first 'counts' for Maariv, while the "Tashlumin" is always the repeated second one.