  The spin-$1/2$ Heisenberg chain with a ferromagnetic first-neighbor exchange
coupling $J_1$ and an antiferromagnetic second-neighbor $J_2$ has a Haldane
dimer ground state with an extremely small spin gap. Thus, the ground state is
readily altered by perturbations. Here, we investigate the effects of XXZ
exchange magnetic anisotropy of both the easy-axis and easy-plane types and an
alternation in $J_1$ on the ground state, the spin gap, and magnetic properties
of the frustrated ferromagnetic spin-$1/2$ chain. It is found that there are
two distinct dimerized spin-gap phases, in one of which the spin gap and the
magnetic susceptibility are extremely small around the SU(2) symmetric case and
in the other they are moderately large far away from the SU(2) symmetric case.
A small alternation in the amplitude of $J_1$ rapidly shortens the pitch of
spin correlations towards the four-spin periodicity, as in the limit of
$J_1/J_2\to0$. These effects are not sufficient to quantitatively explain
overall experimentally observed magnetic properties in the
quasi-one-dimensional spin-gapped magnetoelectric cuprate
Rb$_2$Cu$_2$Mo$_3$O$_{12}$ that exhibits ferroelectricity stabilized by a
magnetic field. Our results are also relevant to Cs$_2$Cu$_2$Mo$_3$O$_{12}$,
where the ferromagnetic intrachain and antiferromagnetic interchain order has
recently been found, in a single chain level. We also reveal the nature of
symmetry-protected topological phase transitions in the model by mapping onto
effective spin-1 chain models.
