  Applying the vacuum-to-$B$-meson correlation functions with an interpolating
current for the light vector meson we construct the light-cone sum rules (LCSR)
for the "effective" form factors $\xi_{\parallel}(n \cdot p)$, $\xi_{\perp}(n
\cdot p)$, $\Xi_{\parallel}(\tau, n \cdot p)$ and $\Xi_{\perp}(\tau, n \cdot
p)$, defined by the corresponding hadronic matrix elements in soft-collinear
effective theory (SCET), entering the leading-power factorization formulae for
QCD form factors responsible for $B \to V \ell \bar \nu_{\ell}$ and $B \to V
\ell \bar \ell$ decays at large hadronic recoil at next-to-leading-order in
QCD. The light-quark mass effect for the local SCET form factors
$\xi_{\parallel}(n \cdot p)$ and $\xi_{\perp}(n \cdot p)$ is also computed from
the LCSR method with the $B$-meson light-cone distribution amplitude
$\phi_B^{+}(\omega, \mu)$ at ${\cal O}(\alpha_s)$. Furthermore, the subleading
power corrections to $B \to V$ form factors from the higher-twist $B$-meson
light-cone distribution amplitudes are also computed with the same method at
tree level up to the twist-six accuracy. Having at our disposal the LCSR
predictions for $B \to V$ form factors, we further perform new determinations
of the CKM matrix element $|V_{ub}|$ from the semileptonic $B \to \rho \, \ell
\, \bar \nu_{\ell}$ and $B \to \omega \, \ell \, \bar \nu_{\ell}$ decays, and
predict the normalized differential branching fractions and the $q^2$-binned
$K^{\ast}$ longitudinal polarization fractions of the exclusive rare $B \to
K^{\ast} \, \nu_{\ell} \, \bar \nu_{\ell}$ decays.
