Let $G_n$ be a graph obtained by the strong product of $P_2$ and $C_n$, where $n\geqslant3$. In this paper, explicit expressions for the Kirchhoff index, multiplicative degree-Kirchhoff index and number of spanning trees of $G_n$ are determined, respectively. It is surprising to find that the Kirchhoff (resp. multiplicative degree-Kirchhoff) index of $G_n$ is almost one-sixth of its Wiener (resp. Gutman) index. Moreover, let $\mathcal{G}^r_n$ be the set of subgraphs obtained from $G_n$ by deleting any $r$ vertical edges of $G_n$, where $0\leqslant r\leqslant n$. Explicit formulas for the Kirchhoff index and the number of spanning trees for any graph $G^r_n\in \mathcal{G}^r_{n}$ are completely established, respectively. Finally, it is interesting to see that the Kirchhoff index of $G^r_n$ is almost one-sixth of its Wiener index.