Although aerodynamic shape optimization (ASO) methods have greatly benefited from the development of computational fluid mechanics (CFD) and the progress in high performance computing, their use in industry still poses difficulties related to the shape parameterization which controls the problem dimensionality, and the mesh management which controls the CFD solver robustness and accuracy. In the light of the latest progress in ASO, this work concurrently explores the effect of 4 measures to tackle those challenges: 1) multi-fidelity surrogate model assisted gradient-free optimization, 2) adaptive infill strategies, 3) shape parameterization data reduction and 4) discretization error control. To that end, the constrained multi-objective optimization of a low-Reynolds number compressor blade designed by DLR is considered in multiple optimization configurations each representative of state of the art practices. The CFD solver WOLF and its feature-based mesh adaptation routine is used to provide low- and high-fidelity solutions, respectively obtained with coarse meshes and the RANS Spalart-Allmaras model, and with error-free meshes and the RANS Spalart-Allmaras BCM model, which accounts for laminar/turbulent transition occurring on the blade. Starting off from a baseline geometry perturbed with free-form deformation (FFD) and 8 control points, the linear principal component analysis data reduction techniques is used to reduce the problem dimensionality. Afterwards, surrogate assisted optimization is performed using Bayesian (autregressive co-kriging) and non-Bayesian (multi-fidelity deep neural network) multi-fidelity surrogates, with their associated Bayesian and non-Bayesian infill strategies. Reference results of an optimization performed using high-fidelity simulations only are used to evaluate the effect of each improvement factor in terms of produced optimal designs and speed of convergence. Benefits and limitations of each of them are discussed and lessons learned are exposed through the prism of extreme multi-fidelity evaluation cost imbalance such as in LES/RANS multi-fidelity conditions.