Are neutrinos and sterile neutrinos both dark matter candidates? Are both neutrinos and sterile neutrinos candidates for dark matter?
In particular, why would "standard" neutrinos be a candidate for dark matter, since they interact with matter?
Why would explicitly sterile neutrinos be needed, since standard neutrinos already interact very weakly?
Why are neutrinos and sterile neutrinos considered as separate dark matter candidates?
 A: 
In particular, why would "standard" neutrino be a candidate to dark matter, since it interacts with matter ?

From $\beta\beta$ decays, we find an upper limit for the density parameter for neutrinos of $0.02\ll\Omega_\text{DM}$; therefore, neutrinos can only account for a small part of the total dark matter component.
Furthermore, due to the neutrino's mass being significantly smaller than its decoupling temperature, neutrinos are relativistic at the time of decoupling (forming so-called "hot" dark matter). Hot dark matter is consistent with a top-down evolution of the Universe, in which small structures are rapidly washed out while larger structures form first. Our Universe seems compatible with the opposite regime, that of bottom-up, or hierarchical, formation, whereby small structures are built first. Hence why our best model is currently ΛCDM -- CDM stands for cold dark matter.
Sterile neutrinos with keV scales are also relativistic upon decoupling and are warm dark matter candidates.
UPDATE Upon re-reading your question, I feel I need to stress the difference between hot dark matter (e.g. neutrinos) and warm dark matter WDM (e.g. sterile neutrinos). WDM particles are relativistic upon decoupling (just like HDM), but are non-relativistic by the time of matter-radiation equality (when the energy densities of matter and radiation are equal). These two categories have different implications for structure formation histories. As I mentioned, hot dark matter does not appear compatible with observations. Warm dark matter is instead consistent with bottom-up formation, as the WDM particles do not cluster, but "stream freely".
