Computing a subproduct tree
How does the computation scale?

Keep reading!  The paper describes how this scales for $n$, in the very next line after the excerpt you show.  In particular, the next line is Proposition 1, which states the running time of computing the subproduct tree, as a function of $n$.  That tells you how the running time scales for large $n$.

How do you compute the coefficients?

Follow the algorithm described in the paper!  You already described the algorithm.  Use exactly what you described.

Don't the coefficients become large, when $n$ is large?

Yes, potentially.  This is inherent to the task: if you want to compute some polynomial (the product) whose coefficients are large, then yes, the coefficients produced by such an algorithm will be large.  This is pretty much a tautological statement.