  A frustrated spin-$1/2$ XXZ chain model comprising a ferromagnetic
nearest-neighbor coupling with the bond alternation, $J_1(1\pm\delta)<0$, and
an antiferromagnetic second-neighbor exchange coupling $J_2>0$ is studied at
zero and weak magnetic fields by means of density matrix renormalization group
calculations of order parameters, correlation functions and the entanglement
entropy as well as an Abelian bosonization analysis. At zero magnetic field,
the bond alternation $\delta>0$ suppresses the gapless phase characterized by a
vector-chiral (VC) long-range order (LRO) and a quasi-LRO of an incommensurate
spin spiral, whereas this phase occupies a large region in the space of
$J_1/J_2$ and the easy-plane exchange anisotropy for $\delta=0$ [S. Furukawa
\textit{et al.}, Phys. Rev. Lett. \textbf{105}, 257205 (2010)]. Then, four
gapped phases are found to appear as the exchange anisotropy varies from the
SU(2)-symmetric case to the U(1)-symmetric case; the Haldane dimer (D$_+$)
phase with the same sign of the $x,y$- and $z$-component dimer order
parameters, two VC dimer (VCD$_+$/VCD$_-$) phases with the sign of the
$z$-component dimer order parameter being unaltered/reversed, and the
even-parity dimer (D$_-$) phase. At small magnetic fields, a field-induced
ring-exchange interaction, which is proportional to a staggered scalar
chirality and a magnetic flux penetrating the associated triangle, drives a
transition from the D$_-$ phase into a VC-Neel-dimer (VCND) phase, but not from
the D$_+$ phase. This VCND phase is stable up to the large magnetic field at
which the Zeeman term closes the spin gap. A possible relevance to
Rb$_2$Cu$_2$Mo$_3$O$_{12}$ is discussed.
