  We study the merger of black hole (BH)-neutron star (NS) binaries with a
variety of BH spins aligned or anti-aligned with the orbital angular momentum,
and with the mass ratio in the range MBH/MNS = 2--5, where MBH and MNS are the
mass of the BH and NS, respectively. We model NS matter by systematically
parametrized piecewise polytropic equations of state. The initial condition is
computed in the puncture framework adopting an isolated horizon framework to
estimate the BH spin and assuming an irrotational velocity field for the fluid
inside the NS. Dynamical simulations are performed in full general relativity
by an adaptive mesh refinement code, SACRA. The treatment of hydrodynamic
equations and estimation of the disk mass are improved. We find that the NS is
tidally disrupted irrespective of the mass ratio when the BH has a moderately
large prograde spin, whereas only binaries with low mass ratios, MBH/MNS <~ 3
or small compactnesses of the NSs, bring the tidal disruption when the BH spin
is zero or retrograde. The mass of the remnant disk is accordingly large as >~
0.1 Msun, which is required by central engines of short gamma-ray bursts, if
the BH spin is prograde. Information of the tidal disruption is reflected in a
clear relation between the compactness of the NS and an appropriately defined
"cutoff frequency" in the gravitational-wave spectrum, above which the spectrum
damps exponentially. We find that the tidal disruption of the NS and excitation
of the quasinormal mode of the remnant BH occur in a compatible manner in high
mass-ratio binaries with the prograde BH spin. The correlation between the
compactness and the cutoff frequency still holds for such cases. It is also
suggested by extrapolation that the merger of an extremely spinning BH and an
irrotational NS binary does not lead to formation of an overspinning BH.
