  We study possible systematic effects on the values of the cosmological
parameters measured through strong lensing analyses of the HFF 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 estimated time delays. Starting from our reference model
in a flat $\Lambda$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 of $H_{0}$. We perform
true blind tests by considering a range of time delays around its original
estimate of $345 \pm 10$ days, as an accurate measurement of this time delay
was 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 of $H_{0}$
and $\Omega_{\rm m}$ 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 for $H_{0}$ and $\Omega_{\rm m}$, respectively. Furthermore, our
model accurately reproduces the extended surface brightness distribution of the
supernova host, covering more than $3 \times 10^{4}$ $HST$ pixels. We also
present the interesting possibility of measuring the value of the
equation-of-state parameter $w$ of the dark energy density, currently with a
30% uncertainty. We conclude that time-delay cluster lenses have the potential
to become soon an alternative and competitive cosmological probe.
