I have couple of questions?
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.
There's lots to your question, so this will only be a partial answer until we clear up some things.
Minimum Phase
A minimum phase filter is one that has all its poles and zeros inside the unit circle.
So now let's partially address your first question:
Is there a relationship between a FIR filter being minimum phase/mixed phase and being causal?
By definition, a minimum phase system is causal.
To answer the second part (about mixed phase filters), we need to understand what causality is.
Causal
A filter is causal if the output does not depend on future inputs.
So now let's address the second part of your first question:
Is there a relationship between a FIR filter being minimum phase/mixed phase and being causal?
If your FIR filter can be written as: $$ H(z) = \sum_{p=-M}^{+N} h_p z^{p} $$ then $H(z)$ is causal if $N \le 0$ (i.e. there are only zero or negative powers of $z$).
Constant Phase
You make the statement:
I understand that a symmetric/non symmetric filter has a constant phase
which is completely incorrect.
The only FIR filters with constant phase are the ones where $N=0$ and $M=0$ so that $H(z) = h_0$ (i.e. a constant).
Symmetric FIR filters and anti-symmetric FIR filters can have constant group delay (and generalized linear phase).
Group Delay
The group delay of a minimum phase FIR filter will be a function of frequency, because minimum phase FIR filters cannot have constant group delay (except in the trivial case of $H(z) = h_0$ noted above.
Placement of zeros
I get filter which has zeros outside the unit circle, and impulse response which is non symmetric. I am told that this is typical of non-causal filter.
It is perfectly possible for a causal FIR filter to have zeroes outside the unit circle.
Break
I think your real question at the bottom is about system identification, but I needed to clarify the misconceptions in your build up first.
Let me know if this makes sense, and I'll try to address the rest of your question later.
