  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.
