Thanks for the questions.
1. Finding a digit sum is based on ‘By Addition’, Sutra 7, and casting out 9s is based on Sutra 5: ‘If the Total is the Same it is Zero’.
2. You will see more applications. It can be used in algebra too and for finding a missing digit or coefficient.
3. Patterns are always useful as they lead to more general concepts and ways of extending them and seeing connections with other patterns. It has been said that mathematics is all about the study of patterns. Fot the Vedic Square designs I am not aware of applications other than the fun in generating patterns: someone should look into this 🙂