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Neutrino emission from gamma-ray bursts (GRBs) has been sought for a long time, and stringent limits on the most accredited GRB emission models have been obtained from IceCube. Multi-wavelength GRB observations of the last decades improved our knowledge of the GRB emission parameters, such as the Lorentz factor and the luminosity, which can vary from one GRB to another by several orders of magnitude. Empirical correlations among such parameters have been identified during the prompt phase, with direct implications on GRB models. In this work, we use the PSLab open-access code, developed for IceCube data analyses, to search for individual neutrino emission from the prompt and afterglow phases of selected GRBs, and for stacking emission from the ensemble of such GRBs. For the afterglow phase, we focus in particular on GRBs with X-ray flares and plateaus. While past stacking searches assumed the same GRB fluence at Earth, we present a stacking scheme based on physically motivated GRB weights. Moreover, we conceive a new methodology for the prompt phase that uses the empirical correlations to infer the GRB luminosity and Lorentz factor, when redshift measurements are not available. We do not observe any significant neutrino excess. Hence, we set constraints on the GRB neutrino fluxes and on relevant GRB parameters, including the magnetic field in the jet. Notably, the baryon loading is found to be <10 for typical GRB prompts, thus disfavoring a baryonic-dominated origin of the GRB ejecta.