Algebraic FEM-FCT and FEM-TVD schemes are integrated into incompressible flow solvers based on the `Multilevel Pressure Schur Complement` (MPSC) approach. It is shown that algebraic flux correction is feasible for nonconforming( rotated bilinear)  finite element approximations on unstructured meshes. Both (approximate) operator-splitting and fully coupled solution strategies are introduced for the discretized Navier-Stokes equations. The need for development of robust and efficient iterative solvers (outer Newton-like schemes, linear multigrid techniques, optimal smoothers/preconditioners) for implicit high-resolution schemes is emphasized. Numerical treatment of extensions (Boussinesq approximation, k-epsilon turbulence model) is addressed and pertinent implementation details are given. Simulation results are presented for three-dimensional benchmark problems as well as for prototypical applications including multiphase and granular flows.