  A new method for estimating the parameter ensuring the fulfillment of the
Mermin-Wagner theorem in the strong coupling diagram technique (SCDT) for the
two-dimensional Hubbard model is suggested. With the precise parameter value,
calculated magnetic quantities are in good agreement with the results of
numeric and optical-lattice experiments. Obtained spin and charge vertices are
used for investigating superconductivity in the $t$-$U$ and $t$-$t'$-$t"$-$U$
Hubbard models in the regime of strong correlations. We found no
superconducting transition in the $t$-$U$ model. In the $t$-$t'$-$t"$-$U$
model, the transition occurs for the singlet $d_{x^2-y^2}$ pairing at
$T_c\approx0.016t$. The difference between the two models is in the
renormalized hopping describing electron motion in SCDT. In the $t$-$U$ model,
it vanishes for momenta $(\pi,0)$, $(0,\pi)$ of extrema of the $d$-wave order
parameter, while the hopping is finite in the $t$-$t'$-$t"$-$U$ model. In the
one-band model, there are optimal values of $t'$ and $t"$ ensuring the highest
$T_c$.
