  Several conceptual aspects of quantum gravity are studied on the example of
the homogeneous isotropic LQC model. In particular: $(i)$ The proper time of
the co-moving observers is showed to be a quantum operator {and} a quantum
spacetime metric tensor operator is derived. $(ii)$ Solutions of the quantum
scalar constraint for two different choices of the lapse function are compared
and contrasted. In particular it is shown that in case of model with masless
scalar field and cosmological constant $\Lambda$ the physical Hilbert spaces
constructed for two choices of lapse are the same for $\Lambda<0$ while they
are significantly different for $\Lambda>0$. $(iii)$ The mechanism of the
singularity avoidance is analyzed via detailed studies of an energy density
operator, whose essential spectrum was shown to be an interval $[0,\rhoc]$,
where $\rhoc\approx 0.41\rho_{\Pl}$. $(iv)$ The relation between the
kinematical and the physical quantum geometry is discussed on the level of
relation between observables.
