  The $\Upsilon(3S)$ production and polarization at high energies is studied in
the framework of $k_T$-factorization approach. Our consideration is based on
the non-relativistic QCD formalism for bound states formation and off-shell
production amplitudes for hard partonic subprocesses. The transverse momentum
dependent (TMD, or unintegrated) gluon densities in a proton were derived from
the Ciafaloni-Catani-Fiorani-Marchesini (CCFM) evolution equation as well as
from the Kimber-Martin-Ryskin (KMR) prescription. Treating the non-perturbative
color octet transitions in terms of the multipole radiation theory and taking
into account feed-down contributions from radiative $\chi_b(3P)$ decays, we
extract the corresponding non-perturbative matrix elements for $\Upsilon(3S)$
and $\chi_b(3P)$ mesons from a combined fit to $\Upsilon(3S)$ transverse
momenta distributions measured by the CMS and ATLAS Collaborations at the~LHC
energies $\sqrt s = 7$ and $13$~TeV and central rapidities. Then we apply the
extracted values to describe the CDF and LHCb data on $\Upsilon(3S)$ production
and to investigate the polarization parameters $\lambda_\theta$, $\lambda_\phi$
and $\lambda_{\theta\phi}$, %and $\tilde\lambda$, which determine the
$\Upsilon(3S)$ spin density matrix. Our predictions have a good agreement with
the currently available data within the theoretical and experimental
uncertainties.
