Is it permissible to offer prayer at home, or should it be offered in the mosque?
All perfect praise be to Allah, The Lord of The Worlds. It isn`t preferable for the man to pray at home as praying in the mosque is twenty seven times more rewarding. Therefore, this should motivate him to offer prayers in the mosque. And Allah Knows Best.
Should missed Sunnah prayer be made up?
Making up missed voluntary acts of worship is from Sunnah.
Should a minor fulfill his vow?
The vow of a minor is countless because he isn`t required to meet the Sharia rulings at such an age, and it is desirable for him to fulfill that vow once he reaches puberty.
Is it permissible to combine Zuhr and Asr prayers for being occupied with a wedding ceremony?
No, it isn`t permissible to combine Zuhr and Asr, or Maghrib and Isha because of being busy with a wedding since the exemption for combining prayers is based on lawful excuses, and this isn`t one of them. And Allah Knows Best.