[Viktor2]

Entropy as we know is the degree of disorder or randomness which we see changing even in case of a system where ΔH is zero e.g., in case of mixing of two gases.
When we define it mathematically we say dS = δq_rev /T

My first question is how can we explain both together. As in the first case we see the entropy changing even in case of δq = 0 but the second part says if δq is zero dS = 0.

My second question is that is it compulsory for an adiabatic process to be reversible to say it isentropic. As we see that being adiabatic only makes δq = 0.

[Corribus]

When we define it mathematically we say dS = δq_rev /T

The full Clausius expression is actually an inequality, not an equation. It should be

dS ≥ dq/T

The equality only applies for a reversible process (which you note in your equation, but forgot about apparently ). A free expansion of an ideal gas, for example, is not a reversible process, hence the equality does not apply.

http://en.wikipedia.org/wiki/Free_expansion

So here even if the heat exchange is zero, the entropy change can still be (and is) positive.