I used dual time integrator (backward-euler / euler method).
Then i compared time step effect using 3 time steps.

If three results was scaled in time, the results were (almost) the same. Therefore, it seemed like the result of a steady state simulation.
Is it correct that the backward-euler(euler) method is working in unsteady simulation?

Euler methods aren’t very accurate and this can impact there convergence. A good paper that considers this is by Lai Wang et al. https://arxiv.org/pdf/1904.04825.pdf

Ultimately, if you perform a step forwards you what to get the most for your money and for incompressible Navier–Stokes using dual time stepping, that often means using a DIRK scheme. Euler is good to have as it very dissipative which can be very useful in testing or, as you say, initialisation. Or I know in more complex physics, say multi-phase with mateirals etc, they like it due to it’s dissipation.

When using dual time stepping dt is the physical time step and pseudo-dt is the pseudo time step. When usinf the standard time integrator there is only dt and this is the physical time step.