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). So there is not much limit on the answer to question 1.

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). So there is not much limit on the answer to question 1.

Also all groups of the same square-free order are Sylow-isomorphic.