Question 1(what is the group for the Shimura datum): 

Well, remember that $H^\times$ is just a bare group. A Shimura datum requires an algebraic group over $\mathbf{Q}$: that is, a functor from $\mathbf{Q}$-schemes to groups. Assuming you mean the group whose $\mathbf{Q}$ points, yes and you can see this in example 5.24 of http://www.jmilne.org/math/xnotes/svi.pdf although you can also use the algebraic group whose $\mathbf{Q}$ points are the *norm one* units of $H$ (which is another example in milne's notes).

Question 2(what is the map from this group to the symplectic group): 

I don't know. Is it even clear that a forgetful map of coarse moduli spaces which happen to be Shimura varieties induces a morphism of Shimura data? Either way your question sounds strongly related to the work of [Victor Rotger's thesis][1] which asks about the irreducibility of the quaternionic locus in $\mathcal{A}_2$.


  [1]: http://www-ma2.upc.edu/vrotger/publicacions_en.html#