This study enables the use of explicit and compact high-order finite-difference schemes with a line-based solver to solve conservation laws on unstructured grids. A quadrilateral subdivision process is used to identify unique line structures (also known as Hamiltonian loops) along which stencil-based discretization techniques are used. To demonstrate the methodology for canonical flows represented by the compressible Navier-Stokes equations, finite-difference explicit and compact spatial discretization up to sixth-order with a maximum of tenth-order low-pass filters as well as finite-volume MUSCL and WENO5 spatial reconstruction schemes are implemented along the Hamiltonian loops. The filter restores the benefits of the high-order approach even in the presence of grid discontinuities that cause changes in loop curvature. These schemes are then extended for the simulation of high-speed flows to capture shocks. Two strategies were explored; an ``adaptive filter" approach, which combines the finite-difference scheme with a locally reduced-order filter and a hybrid finite-difference/finite-volume approach. It is shown that finite-difference is effective and accurate for smooth regions, while the finite-volume is robust in capturing discontinuities and shocks. Two distinct shock indicators are used, one based on pressure gradients and the other on divergence of velocity. To achieve faster convergence, spatial discretization is applied with line-based Alternating Direction Implicit (ADI) time integration. A key focus of this work is to demonstrate the proposed methodology on flow scenarios that test various aspects of the methodology, and consequently, the cases studied are Isentropic vortex convection, lid-driven cavity, double periodic shear layer, viscous and inviscid flow past a cylinder, stationary normal shock, subsonic and transonic inviscid flow past a NACA 0012 airfoil, supersonic cylinder, unsteady flow past tandem circular cylinders, and supersonic flow past a forward-facing step. The ability of some schemes, particularly the fourth-order explicit, to nearly achieve the formal order of accuracy and successfully predict the flow physics is one of the key findings. The applicability of the hybrid finite-difference/finite-volume approach performed well with excellent agreement against available results, sometimes with far fewer degrees of freedom.