For comment #2 I would suggest looking into the immersed boundary (IB) method. The idea behind this method is to combine an Eulerian description of the fluid with a Lagrangian description of the solid structure. The fluid-structure coupling is achieved by injecting a force term into the Navier-Stokes momentum equations:
$$
\rho\left(\frac{\partial \vec{u}(\vec{x},t)}{\partial t} + \vec{u} \cdot \nabla \vec{u}\right) = - \nabla p + \mu \Delta \vec{u}(\vec{x},t) + \vec{f}(\vec{x},t)
$$
where $\vec{f}(\vec{x},t)$ is designed to mimic the effect of solid on the fluid.
Imagine a thin immersed structure $\Gamma$, parameterized by the curve $\vec{X}(s,t)$ with respect to a Lagrangian coordinate $s$, and under the influence of a net force $\vec{F}(s,t)$. The force that will be exerted back on the fluid is given as:
$$
f(\vec{x},t) = \int_\Gamma \vec{F}(s,t) \delta(\vec{x} - \vec{X}(s,t)) ds,
$$
where $\delta$ is the Dirac $\delta$ function. Essentially, this equation says the Lagrangian boundary force is spread to the fluid. In three-dimensions the parameterization is over a surface, and the force-spreading is a surface integral.
At the same time, the no-slip boundary condition implies the points on the surface structure $\vec{X}(s,t)$ and the ambient fluid at position $X$ move with the same velocity:
$$
\dot{\vec{X}}(t) = \frac{\partial\vec{X}(t)}{\partial t} = \vec{u}(\vec{X}(t),t)
$$
This can be rewritten using the Dirac delta as:
$$
\dot{\vec{X}}(t) = \int \vec{u}(\vec{x},t) \delta(\vec{x} - \vec{X}) d^3 x
$$
Since for the numerical solution (FVM) the velocity values are only known at the Eulerian grid-points, the Dirac delta functions are replaced with discrete kernel functions. In the discrete formulation, the equation above defines a velocity-interpolation. Through the steps of force-spreading and velocity interpolation, a bi-directional coupling is achieved between solid and fluid.
The forces $\vec{F}(s,t)$ acting trough the solid structure can be designed to reproduce rigid bodies, elastic membranes, or even objects such as red-blood cells which follow more complex constitutive equations. The original purpose of the method was for simulating blood flow in the heart.
With respect to comment #1 I might just add the remark, that due to limited grid resolution, your IB objects might behave as if having a small non-zero thickness. It is then a matter of model validation, to see if the errors are acceptable.
To learn more I can highly recommend the following short tutorial written by Timm Krüger: Basics of the immersed boundary method, and also a lengthier article from the inventor of the method - Charles S. Peskin - The immersed boundary method. Some numerical codes are linked on the Wikipedia page. The method also combines well with the lattice Boltzmann method, if that is an option for you.
