I report some definitions of flow characteristics as given on Cengel Cimbala.

• Laminar or turbulent

Laminar flow: A stable well-ordered state of fluid flow in which all pairs of adjacent fluid particles move alongside one another forming laminates. A flow that is not laminar is either turbulent or transitional to turbulence, which occurs above a critical Reynolds number.

Turbulent flow: An unstable disordered state of vortical fluid flow that is inherently unsteady and that contains eddying motions over a wide range of sizes (or scales). Turbulent flows are always at Reynolds numbers above a critical value that is large relative to 1. Mixing is hugely enhanced, surface shear stresses are much higher, and head loss is greatly increased in turbulent flows as compared to corresponding laminar flows.

Steady flow : A flow in which all fluid variables (velocity, pressure, density, temperature, etc.) at all fixed points in the flow are constant in time (but generally vary from place to place). Thus, in steady flows all partial derivatives in time are zero. Flows that are not precisely steady but that change sufficiently slowly in time to neglect time derivative terms with relatively little error are called quasi-steady.

Unsteady flow: A flow in which at least one variable at a fixed point in the flow changes with time. Thus, in unsteady flows a partial derivative in time is non-zero for at least one point in the flow.

• Are these two couple of characteristics completely independent one from the other? In other words, can I have, for instance, besides a laminar steady flow and a turbulent unsteady flow, also a laminar unsteady flow or a turbulent steady flow? (Which sounds very strange)
• Can an ideal fluid ($\rho=\textrm{const}$ and $\eta=0$), in steady flow (for which Bernoulli equation is valid) be in a turbulent flow? Or is it implicitly considered to be in laminar flow?

Either kind of flow can exhibit steady flow or unsteady flow. Unsteady flow is where, over a large time scale, things are changing at each spatial location with time. When they say that turbulent flow is inherently unsteady, what they mean is that, over a small time scale (at each spatial location), the velocity components are varying rapidly with time, but, when averaged over relatively small time intervals, the average velocity components vary much more slowly, or not at all. If they do not vary at all, the turbulent flow is considered steady. If the vary slowly (at each spatial location) the turbulent flow is considered unsteady.

Regarding inviscid flow, it is assumed that viscous and turbulent stresses are both considered insignificant. So it is considered neither laminar nor turbulent. It would be what you would obtain if the fluid had a very low viscosity, but the flow could not transition to turbulent flow.

You can have all the cases except steady turbulent flow. Of course you can have $\textit{statistically}$ steady turbulent flow, which is to say that averages can be steady in time (refer @Chester Miller's answer).

Turbulent flow is by definition something that is highly vortical and dissipative. In absence of other mechanisms, viscosity is the only way to generate vorticity (for e.g. in a boundary layer over a solid boundary). Also dissipation of kinetic energy of turbulence requires viscosity again. In an inviscid fluid, due to absence of viscosity, there is neither vorticity generation (it only contains what was given to it initially, read up Kelvin's circulation theorem) nor dissipation of mechanical energy contained in turbulence. Such a flow, no matter how complicated, cannot be called turbulent, as turbulence is defined presently.