How does this relationship between the Catalan numbers and SU(2) generalize?

The Catalan numbers enumerate (amongst everything else!) the bases of the Temperley-Lieb algebras. These algebras $TL_n(q)$ are exactly $\operatorname{End}_{U_q \mathfrak{su}_2}(V^{\otimes n})$ where $V$ is the standard representation.

If $q$ is a $2k+4$-th root of unity, the semisimplified representation theory of $U_q \mathfrak{su}_2$ becomes '$A_{k+1}$': that is, it has $k+1$ simples, and the principal graph for tensor product with the standart is $A_{k+1}$. The algebra $\operatorname{End}_{U_q \mathfrak{su}_2}(V^{\otimes n})$ is now smaller (some of the summands of the tensor power cancel out when you use the quantum Racah rule), and in fact it is enumerated by loops on $A_{k+1}$ based at the first vertex. You can pick out a subset of the entire Temperley-Lieb that gives a basis: just use your description of paths as trees, and take the dual graph.

Since we're looking at a unitary tensor category, the dimension of the standard object is exactly the Frobenius-Perron eigenvalue (largest real eigenvalue of the adjacency matrix) of the principal graph.

Re Question 1: For the connection between walks and continued fractions (which is closely related to orthogonal polynomials, but not, as far as I know, to root systems) see Philippe Flajolet, Combinatorial Aspects of Continued Fractions, Discrete Mathematics 32 (1980), pp. 125–161. Flajolet's paper has been very influential and many people have done further work in this direction.

For Question 1, you might want to consult Viennot's monograph Une theorie combinatoire des polynomes orthogonaux.