The Answer to Question 1 is YES. It follows from Serre's variant of Hilbert's irreducibility theorem (for infinite Galois extensions) combined with the Tate conjecture on homomorphisms of abelian varieties in char 0 (proven by Faltings); see Prop. 1.3 and Cor. 1.5 of 1995 Compositio paper by Rutger Noot (vol 97, Oort Festschrift) for details.
The main idea is as follows. If $A$ is an abelian variety over a finitely generated field $K$ of char 0 and $\ell$ is a prime then there is an abelian variety $A_0$ over a number field $K_0$ that is a ``specialization/reduction" of $A$ and such that a canonical (up to a "conjugation") isomorphism of Tate modules $T_{\ell}(A)$ and $T_{\ell}(A_0)$ induces an isomorphism between the corresponding $\ell$-adic images of the absolute Galois groups of $K$ and $K_0$ in the automorphism groups of $T_{\ell}(A)$ and $T_{\ell}(A_0)$ respectively. Since $End(A)$ embeds into $End(A_0)$, one obtains that the endomorphism rings of $A$ and $A_0$ are isomorphic.
