How is it possible that Saturn's gravitational acceleration felt by Mimas is stronger than Mimas' own surface gravity? The surface gravity on Mimas is $≈ 0.063\text{ m}/\text{s}^2$ and Saturn's gravitational acceleration at the distance of Mimas' orbit is:
$$\frac{{GM}}{{r}^2} = \frac{{6.674 \times 10^{-11} \times 568.34 \times 10^{24}}}{{(185.52 \times 10^{6})}^2} ≈  1.102 \text{ m}/\text{s}^2$$
How can this be? An object on Mimas' surface would be much more attracted to Saturn than it is to Mimas. Shouldn't Mimas itself get ripped apart or is my math wrong?
An object on Mimas' surface would be much more attracted to Saturn than it is to Mimas.

You are missing that Mimas as a whole accelerates gravitationally toward Saturn. What this means is that a point on the surface of the Mimas will feel the acceleration at that point toward Saturn minus the acceleration of Mimas as a whole toward Saturn. This is the tidal acceleration. It is equal to
$$a_\text{tidal} = \left|\frac{GM}{(R\pm r)^2}-\frac{GM}{r^2}\right| \approx 2 \frac{GMr}{R^3} = 2\frac{GM}{R^2}\frac{r}{R}$$
where $R$ is Mimas' semi major axis length and $r$ is Mimas' mean radius. The approximation assumes that $r\ll R$, which most certainly is the case given that Mimas's radius is about 1/1000 of the semi-major axis length of its orbit about Saturn. The result is rather small, about 0.002355 m/s2.