Investigating cosmic distance duality and dark energy evolution through intermediate and high-z probes

We investigate deviations from the cosmic distance duality relation adopting model-dependent and -independent approaches using i) a Taylor expansion, ii) a power-law parameterization, iii) a logarithmic correction, iv) a (2;1) Pad\'e polynomial and v) a second order Chebyshev parameterization. We derive constraints on all parameters using observational Hubble data, galaxy clusters, type Ia supernovae, DESI data and gamma-ray bursts. Through Monte-Carlo Markov chain analyses adopting the Metropolis Hastings algorithm, we find no significant violation of duality, then model selection criteria favor flat scenarios even though a slight curvature is not totally ruled out. For the $H_0$ tension we find a preference at $1$-$\sigma$ for $h^R_0=0.730\pm0.010$ from supernovae when dropping DESI data and for $h^P_0=0.674\pm 0.005$ from Planck when using DESI and gamma-ray bursts.