Let $G_1$ and $G_2$ be nonisomorphic Sylow-isomorphic groups. For example let $G_1 = C_6$ and $G_2 = S_3$. Then for any finite group $H$, the groups $G_1 \times H$ and $G_2 \times H$ are nonisomorphic Sylow-isomorphic groups (by [Krull--Schmidt][1]). So there is not much limit on the answer to question 1.


  [1]: https://en.wikipedia.org/wiki/Krull%E2%80%93Schmidt_theorem