  We consider the chemotaxis-Navier-Stokes system with generalized fluid
dissipation in $\mathbb{R}^3$:
  \begin{eqnarray*}
  \begin{cases} \partial_t n+u\cdot \nabla n=\Delta n- \nabla \cdot (\chi(c)n
\nabla c),\\ \partial_t c+u \cdot \nabla c=\Delta c-nf(c),\\ \partial_t u +u
\cdot \nabla u+\nabla P=-(-\Delta)^\alpha u-n\nabla \phi,\\ \nabla \cdot u=0,
  \end{cases} \end{eqnarray*} which describes the motion of swimming bacteria
or bacillus subtilis suspended to water flows. First, we prove some blow-up
criteria of strong solutions to the Cauchy problem, including the Prodi-Serrin
type criterion ($\alpha>\frac{3}{4}$) and the Beir${\rm\tilde{a}}$o da Veiga
type criterion $(\alpha>\frac{1}{2})$. Then, we verify the global existence and
uniqueness of strong solutions for arbitrarily large initial fluid velocity and
bacteria density for $\alpha\geq \frac{5}{4}$. Furthermore, in the scenario of
$\frac{3}{4}<\alpha<\frac{5}{4}$, we establish uniform regularity estimates and
optimal time-decay rates of global solutions if the $L^2$-norm of initial data
is small. To our knowledge, this is the first result concerning the global
existence and large-time behavior of strong solutions for the
chemotaxis-Navier-Stokes equations with possibly large oscillations.
