Several interesting correlations among gamma-ray burst (GRB) observables with available redshifts have been recently identified. Proper evaluation and calibration of these correlations may facilitate the use of GRBs as standard candles constraining the expansion history of the universe up to redshifts of z > 6. Here we use the 69 GRB data set recently compiled by Schaefer and we test the calibration of five of the above correlations [(1) E-peak-E-gamma, (2) E-peak-L, (3) tau(lag)-L, (4) V-L, (5) tau(RT)-L] with respect to two potential sources of systematics: evolution with redshift and cosmological model used in the calibration. In examining the model dependence we assume flat Lambda CDM and vary Omega(m). Our approach avoids the circularity problem of previous studies since we do not fix Omega(m) to find the correlation parameters. Instead we simultaneously minimize chi(2) with respect to both the log - linear correlation parameters a, b and the cosmological parameter Omega(m). We find no statistically significant evidence for redshift dependence of a and b in any of the correlation relations tested [ the slopes of a(z) and b(z) are consistent with 0 at the 2 sigma level]. We also find that one of the five correlation relations tested (Epeak - E.) has a significantly lower intrinsic dispersion compared to the other correlations. For this correlation relation, the maximum likelihood method (chi(2) minimization) leads to b(1) = 50.58 +/- 0.04, a(1) = 1.56 +/- 0.11, Omega(m) = 0.30(-0.25)(+1.60), respectively. The other four correlation relations minimize chi(2) for a flat matter dominated universe Omega(m) similar or equal to 1. Finally, a cross-correlation analysis between the GRBs and type Ia supernova data for various values of Omega(m) has shown that the Epeak - E. relation traces well the SnIa regime (within 0.17 <= z <= 1.755). In particular, for Omega(m) similar or equal to 0.15 and Omega(m) similar or equal to 0.30 we get the highest correlation signal between the two populations. However, due to the large error bars in the cross-correlation analysis (small number statistics) even the tightest correlation relation (E-peak-E-gamma) provides much weaker constraints on Omega(m) than current SnIa data.
