We study possible systematic effects on the values of the cosmological parameters measured through strong lensing analyses of the Hubble Frontier Field galaxy cluster MACS J1149.5+2223. We use the observed positions of a large set of spectroscopically selected multiple images, including those of supernova "Refsdal" with their published time delays. Starting from our reference model in a flat ?CDM cosmology, published in Grillo et al. (2018), we confirm the relevance of the longest measurable time delay, between SX and S1, and an approximately linear relation between its value and that ofH(0). We perform true blind tests by considering a range of time delays around its original estimate of 345 10 days, as an accurate measurement of this time delay is still not known at the time of analysis and writing. We investigate separately the impact of a constant sheet of mass at the cluster redshift, of a power-law profile for the mass density of the cluster main halo and of some scatter in the cluster member scaling relations. Remarkably, we find that these systematic effects do not introduce a significant bias on the inferred values ofH(0)and omega(m)and that the statistical uncertainties dominate the total error budget: a 3% uncertainty on the time delay of image SX translates into approximately 6% and 40% (including both statistical and systematic 1 sigma) uncertainties forH(0)and omega(m), respectively. Furthermore, our model accurately reproduces the extended surface brightness distribution of the supernova host. We also present the interesting possibility of measuring the value of the equation-of-state parameterwof the dark energy density, currently with a 30% uncertainty.