Collision resistant hash function
The birthday attack applies to any hash-function, regardless of its structure, and also to $h_i$.

For your first question, here is a possible hint. Note that


if we can find a collision in $h_i$ then we can find a collision in $h_0$ (by considering the top-most $h_0$, for instance).
if we can find a collision in $h_0$ then, we can find a collision for $h_i$ (by replacing one of the inner-most $h_0$ with it's collision).