We present here a conservative finite volume cut-cell method for moving rigid bodies immersed in incompressible flows, implemented in the open-source software Basilisk. The method is constructed starting from a uniform Cartesian grid, in which we embed a discrete representation of a rigid body that intersects underlying cells to form irregular fluid control volumes, or cut-cells. In each cell, we then discretize the Navier-Stokes equations using a fractional-step projection method and insure that at each time step, the finite volume discretization scheme remains spatially second-order and conservative in cut-cells by carefully computing gradients normal to the embedded boundaries, even in degenerated cases. To avoid stability issues due to the well-documented problem of small cut-cells, we use a simple and efficient flux redistribution technique to extend the range of influence of small cut-cells to their neighboring cells. We also provide a time history to emerged cells through a field value reconstruction in the direction normal to the embedded boundaries. In case of freely moving particles, we simply use an explicit weak fluid-solid coupling strategy. Finally, we robustly extend our conservative finite volume cut-cell method for moving boundaries to adaptive Cartesian tree grids by constructing specific restriction and prolongation operators between two consecutive levels of a tree grid in the vicinity of a cut-cell. We successfully test the method on a series of validation test cases ranging from fixed, moving with a prescribed motion to freely moving 2D cylinders and spheres for a wide range of Reynolds (0 ≤ Re ≤ 1000) and Galileo numbers (0 ≤ Ga ≤ 250). While the method is accurate, conservative, robust and efficient, we show that a low-amplitude pressure noise is generated when using mesh adaptation in the limit case of very high Reynolds numbers.