I have to answer the next questions:
What is the number of complete 1-types of the theory of atomless Boolean algebras?
What is the number of complete 2-types of the theory of atomless Boolean algebras?
I know that the theory of atomless Boolean algebras is countably categorical (has up to isomorphism only one countable model) and therefore has only finitely many n-types for each n. But what next?
Hint: (elaborating a little on Chris Eagle's comment) if a theory is $\omega$-categorical, then its only countable model is necessarily saturated.
Now, if you have any saturated model $M$, then it realizes all $n$-types without parameters for each $n$ and it is strongly homogeneous. This means that not only can you find representatives for each $n$-type, but $n$-types correspond exactly to orbits of $\operatorname{Aut}(M)$ acting on $M^n$.