  We use the state-of-the-art data on cosmic chronometers (CCH) and the
Pantheon+ compilation of supernovae of Type Ia (SNIa) to test the constancy of
the SNIa absolute magnitude, $M$, and the robustness of the cosmological
principle (CP) at $z\lesssim 2$ with a model-agnostic approach. We do so by
reconstructing $M(z)$ and the curvature parameter $\Omega_{k}(z)$ using
Gaussian Processes. Moreover, we use CCH in combination with data on baryon
acoustic oscillations (BAO) from various galaxy surveys (6dFGS, BOSS, eBOSS,
WiggleZ, DES Y3) to measure the sound horizon at the baryon-drag epoch, $r_d$,
from each BAO data point and check their consistency. Given the precision
allowed by the CCH, we find that $M(z)$, $\Omega_k(z)$ and $r_d(z)$ are fully
compatible (at $<68\%$ C.L.) with constant values. This justifies our final
analyses, in which we put constraints on these constant parameters under the
validity of the CP, the metric description of gravity and standard physics in
the vicinity of the stellar objects, but otherwise in a model-independent way.
If we exclude the SNIa contained in the host galaxies employed by SH0ES, our
results read $M=(-19.314^{+0.086}_{-0.108})$ mag, $r_d=(142.3\pm 5.3)$ Mpc and
$\Omega_k=-0.07^{+0.12}_{-0.15}$, with $H_0=(71.5\pm 3.1)$ km/s/Mpc ($68\%$
C.L.). These values are independent from the main data sets involved in the
$H_0$ tension, namely, the cosmic microwave background and the first two rungs
of the cosmic distance ladder. If, instead, we also consider the SNIa in the
host galaxies, calibrated with Cepheids, we measure
$M=(-19.252^{+0.024}_{-0.036})$ mag, $r_d=(141.9^{+5.6}_{-4.9})$ Mpc,
$\Omega_k=-0.10^{+0.12}_{-0.15}$ and $H_0=(74.0^{+0.9}_{-1.0})$ km/s/Mpc.
