The interplay between direct aligning interactions, hydrodynamic interactions and self-propulsion in microswimmers gives rise to a wealth of intriguing non-equilibrium properties and distinctive collective behavior such as complex chaotic flows and unusual rheological behavior. To understand the effect of self-propulsion on hydrodynamic interactions, we analyze a minimal model for a rigid spherical microswimmer and explore the consequences of its extended surface on its flow properties. Then, we explore dynamical behavior of dilute suspensions of magnetic swimmers in an external magnetic field using a kinetic theory framework and characterize the conditions under which hydrodynamic instabilities occur. For sufficiently strong self-propulsion and magnetic field strengths, instabilities in density appear that make an orientationally aligned (polar) phase unstable. These hydrodynamic instabilities persist even for strong magnetic fields. We find that the interplay between the hydrodynamic interactions and the coupling to an external magnetic field leads to emergent spatio-temporal patterns that depend on the type of swimmers.