1. Do I have to memorise the formula for the Binomial Theorem?
For examinations it is provided and for most school tests and examinations it will be given to you.
2. What is the use of using the Binomial Theorem for approximating calculations such as 1.067?
Before the days of calculators it would have been quicker. Nowadays it is used to approximate a binomial with a simpler expression like a quadratic.