  We report on a detailed investigation of the spin-1 $J_1-J_2-J_3$ Heisenberg
model, a frustrated model with nearest-neighbor coupling $J_1$, next-nearest
neighbor coupling $J_2$, and a three site interaction $J_3\left[({\bf
S}_{i-1}\cdot {\bf S}_i)({\bf S}_i\cdot {\bf S}_{i+1})+{\mathrm H.c.}\right]$
previously studied in [Phys. Rev. B 93, 241108(R) (2016)]. Using DMRG and exact
diagonalizations, we show that the phase boundaries between the Haldane phase,
the next-nearest neighbor Haldane phase and the dimerized phase can be very
accurately determined by combining the information deduced from the
dimerization, the ground-state energy, the entanglement spectrum and the Berry
phase. By a careful investigation of the finite-size spectrum, we also show
that the transition between the next-nearest neighbor Haldane phase and the
dimerized phase is in the Ising universality class all along the critical line.
Furthermore, we justify the conformal embedding of the $SU(2)_2$
Wess-Zumino-Witten conformal field theory in terms of a boson and an Ising
field, and we explicitly derive a number of consequences of this embedding for
the spectrum along the $SU(2)_2$ transition line between the Haldane phase and
the dimerized phase. We also show that the solitons along the first-order
transition line between the Haldane phase and the dimerized phase carry a
spin-1/2, while those between different dimerization domains inside the
dimerized phase carry a spin 1. Finally, we show that short-range correlations
change character in the Haldane and dimerized phases through disorder and
Lifshitz lines, as well as through the development of short-range dimer
correlations in the Haldane phase, leading to a remarkably rich phase diagram.
