  We study a diffuse interface model that describes the dynamics of
incompressible two-phase flows with chemotaxis effects. This model also takes
into account some significant mechanisms such as active transport and nonlocal
interactions of Oono's type. The system under investigation couples the
Navier-Stokes equations for the fluid velocity, a convective Cahn-Hilliard
equation with physically relevant singular potential for the phase-field
variable and an advection-diffusion-reaction equation for the nutrient density.
For the initial boundary value problem in a smooth bounded domain
$\Omega\subset \mathbb{R}^2$, we first prove the existence and uniqueness of
global strong solutions that are strictly separated from the pure states $\pm
1$ over time. Then we prove the continuous dependence with respect to the
initial data and source term for the strong solution in energy norms. Finally,
we show the propagation of regularity for weak solutions.
