In this work, we enhance the robustness of the DLR in-house finite-volume code ThetaCOM, which solves the reactive Navier-Stokes equations in the low-Mach number regime on unstructured grids with detailed chemistry and high-order, k-exact spatial discretization developed by Setzwein et alii. While high-order methods reduce computational costs by achieving desired accuracy on coarser grids, they suffer from reduced stability due to their inherently low numerical dissipation. Two critical sources of instability are addressed in this work: vortices traversing open boundaries and strong density gradients due to combustion. To increase the robustness of the solver at open boundaries, we implement an energy-stable open boundary condition based on the work of Dong et al. into ThetaCOM and extend it to variable-density flows. Furthermore, we introduce an under-relaxation factor for the density into the iterative projection scheme of ThetaCOM to counter instabilities caused by strong density gradients. We demonstrate the efficacy of the measures outlined previously with several validation cases. Firstly, the highly vortical wake of a two-dimensional square cylinder is simulated on severely truncated grids, even triggering backflow through the open boundary. The resulting aerodynamic coefficients match literature data closely, while the stability of the scheme is maintained. Secondly, a Large Eddy Simulation (LES) of a turbulent, premixed confined jet flame is performed, thereby demonstrating the applicability and industry-relevance of ThetaCOM in combination with our latest improvements. The results are in good agreement with experimental data and the advantages of the k-exact discretization over a conventional second-order discretization are demonstrated. In conclusion, we extended the capabilities of ThetaCOM and advanced the practicability of high-order schemes through increased robustness. The LES of the confined jet flame marks a significant step towards the proliferation of high-order methods in industry applications such as gas turbine combustors.