Entropy, enthalpy, and reversibility
Your text will best explain reversibility.  The short answer is that a reversible process is one whose direction can be reversed by a microscopic change in external conditions.

For example a gas in a cylinder expanding against a piston will do so reversibly as long as the external pressure and the internal pressure are essentially identical.  Of course for expansion the internal pressure must be infinitesimally greater than the external pressure.

Next, entropy is really only defined for systems at equilibrium.

Next, the equation for the entropy above is not correct except at constant temperature.  In reality it is $dS = dq/T$, which generally must be integrated in order to determine the entropy by this method.

For the second question, there is some confusion here. $\Delta H$ is equal to $q$ only at constant pressure.

In fact there are lots of ways to compute the entropy change in a process, but this space is too small to go into them.  They should be in your textbook.