  In Type-II seesaw model, an electroweak triplet scalar field $\Delta$ with a
non-zero vacuum expectation value (vev) $v_\Delta$ is introduced to facilitate
the generation of small neutrino masses. A non-zero $v_\Delta$ also affects the
W mass through the electroweak $\rho$ parameter, making it to be less than 1 as
predicted by standard model (SM). The component fields in $\Delta$ come along
introduce additional contributions to reduce the SM rare neutrino trident
scattering cross section. These fields also induce new processes not existed in
SM, such as $l_i \to \overline{ l_j} l_k l_l$ and $l_i \to l_j \gamma$. There
are severe constraints on these processes which limit the effects on neutrino
trident scattering and the $\rho$ parameter and therefore the W mass. The newly
measured W mass by CDF makes the central value of $\rho$ parameter to be larger
than 1, even larger than previously expected. Combining neutrinoless double
beta decay, direct neutrino mass and oscillation data, we find a lower limit
for $v_\Delta$ as a function of the triplet scalar mass $m_\Delta$, $v_\Delta >
(6.3 \sim 8.4) \mathrm{eV} (100 \mathrm{GeV}/m_\Delta)$. To have significant
effect on $\rho$ in this model, $v_\Delta$ needs to be in the range of a GeV or
so. However this implies a very small $m_\Delta$ which is ruled out by data. We
conclude that the effect of triplet vev $v_\Delta$ on the W mass can be
neglected. We also find that at 3$\sigma$ level, the deviation of the ratio for
Type-II Seesaw to SM neutrino trident scattering cross section predictions is
reduced to be below 1, but is restricted to be larger than 0.98.
