Sum of independent random variables

Yes, they are normally distributed. This is the Lévy-Cramér theorem.

Yes, and the same holds for Poisson, and for mixtures of Gauss and Poisson. All these are special cases of the general question: if $X_j$ are independent and we know the distribution of their sum, what can be said about the distributions of the $X_j$. This general question is addressed in the book Linnik, Ostrovskii, Decomposition of random variables and vectors, AMS 1977 (translation from the Russian).