Supercongruences involving Apéry-like numbers and binomial coefficients

Let $\{S_n\}$ be the Apéry-like sequence given by $S_n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})(\genfrac{}{}{0ex}{}{2k}{k})(\genfrac{}{}{0ex}{}{2n-2k}{n-k})$. We show that for any odd prime $p$, $\sum _{n=1}^{p-1}\frac{n{S}_n}{8^n}\equiv (1-(-1{)}^{\frac{p-1}{2}})p^2(\text{mod}{p}^3)$. Let $\{Q_n\}$ be the Apéry-like sequence given by $Q_n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})(-8{)}^{n-k}\sum _{r=0}^k{(\genfrac{}{}{0ex}{}{k}{r})}^3$. We establish many congruences concerning $Q_n$. For an odd prime $p$, we also deduce congruences for $\sum _{k=0}^{p-1}{(\genfrac{}{}{0ex}{}{2k}{k})}^3\frac{1}{{64}^k}(\text{mod}{p}^3)$, $\sum _{k=0}^{p-1}{(\genfrac{}{}{0ex}{}{2k}{k})}^3\frac{1}{{64}^k(k+1{)}^2}(\text{mod}{p}^2)$ and $\sum _{k=0}^{p-1}{(\genfrac{}{}{0ex}{}{2k}{k})}^3\frac{1}{{64}^k(2k-1)}(\text{mod}p)$, and pose lots of conjectures on congruences involving binomial coefficients and Apéry-like numbers.