We know that the sum of two independent normal random variables is again a normal random variable. But is the reverse right? If $X$ and $Y$ are independent random variables satisfying $X+Y$~$N(\mu,\sigma^2)$ for some $\mu$ and $\sigma$, can we conclude that both $X$ and $Y$ obey normal distribution? or under some conditions added on $p_X$ and $p_Y$ (the density functions of $X$ and $Y$)?
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).
Yes, they are normally distributed. This is the Lévy-Cramér theorem.