In the construction of the surfaces that you mentioned, instead of a 3-valent tree we could as well use an $n$-valent tree (with $n\geq 3$). Also instead of adding one handle at each vertex we could add any positive number g of handles. Would the surfaces so obtained be homeomorphic to each other independent of choice of g and n ?

@Marc thanks for your suggestion. I will try it out.

Thanks for the answer. In Weingarten's formula all group elements in the integrand are in fundamental representation. But I don't think if I can get any more general results:)

Thanks for the answer. I am a physics student and was considering second integral in context of a kind of generalization of two dimensional Yang Mills. I couldn't find any mathematical references considering such integrals so thought about asking here.