Are reversible adiabatic processes always isentropic?

Yes. For a reversible process, we have the relation \begin{align} dS = \frac{\delta Q}{T} \end{align} and for an adiabatic process, we have (by definition) \begin{align} \delta Q = 0, \end{align} which implies that \begin{align} dS=0. \end{align}

Yes.

From Clausius theorem the following inequality can be deduced: $$\delta Q \le TdS$$ where the equality holds in the reversible case.

So, a reversible adiabatic process is necessarily isentropic, but irreversible adiabatic processes are not so.

To put it in another way, in an irreversible process, according to the above inequality, either entropy changes, or heat must be somehow removed from the system to make it possible to have zero change in entropy. So an irreversible isentropic process can not be adiabatic.