  We study the spin $S=1$ Heisenberg chain, with nearest neighbor, next nearest
neighbor ($\alpha$) and biquadratic ($\beta$) interactions using a combination
of the density matrix renormalization group (DMRG), an analytic variational
matrix product state wavefunction, and non-Abelian bosonization. We study the
effect of frustration ($\alpha>0$) on the Haldane phase with $-1\leq \beta < 1$
which reveals a rich phase diagram. For $-1<\beta<\beta^\ast$, we establish the
existence of a spontaneously dimerized phase for large $\alpha>\alpha_c$,
separated from the Haldane phase by the critical line $\alpha_c(\beta)$ of
second-order phase transitions connected to the Takhtajan--Babudjian integrable
point $\alpha_c(\beta=-1)=0$. In the opposite regime, $\beta>\beta^\ast$, the
transition from the Haldane phase becomes first-order into the next nearest
neighbor (NNN) AKLT phase. Based on field theoretical arguments and DMRG
calculations, we conjecture that these two regimes are separated by a
multicritical point ($\beta^\ast, \alpha^\ast$) of a different universality
class, described by the $SU(2)_4$ Wess--Zumino--Witten critical theory. From
the DMRG calculations we estimate this multicritical point to lie in the range
$-0.2<\beta^\ast<-0.15$ and $0.47<\alpha^\ast < 0.53$. We find that the
dimerized and NNN-AKLT phases are separated by a line of first-order phase
transitions that terminates at the multicritical point. Inside the Haldane
phase, we show the existence of two incommensurate crossovers: the Lifshitz
transition and the disorder transition of the first kind, marking
incommensurate correlations in momentum and real space, respectively. We show
these crossover lines stretch across the entire $(\beta,\alpha)$ phase diagram,
merging into a single incommensurate-to-commensurate transition line for
negative $\beta\lesssim \beta^\ast$ outside the Haldane phase.
