Causal and minimum phase
A minimum-phase system is always causal and stable by definition, so in the case of discrete-time systems with rational transfer functions, all poles and zeros are inside the unit circle of the complex $z$-plane. This is why a minimum-phase system can be inverted by a causal and stable system. Because of this definition, it does not make sense to compare non-causal systems and their phase lag or group delay to the corresponding properties of minimum-phase systems. Minimum-phase systems have the smallest possible phase lag of all causal systems with the same magnitude response.

The group delay is a function of frequency. For symmetric/anti-symmetric impulse responses, the group delay is constant. This is probably what you meant in your question. Such linear-phase (constant group delay) systems can never be minimum-phase (unless their impulse response is just a unit impulse at $n=0$). Minimum-phase systems have a frequency dependent group delay, but at any frequency, their group delay is smaller than the group delay of any other causal system with the same magnitude response.