  Relativistic jets originating from protomagnetar central engines can lead to
long duration gamma-ray bursts (GRBs) and are considered potential sources of
ultrahigh-energy cosmic rays and secondary neutrinos. We explore the
propagation of such jets through a broad range of progenitors, from stars which
have shed their envelopes to supergiants which have not. We use a
semi-analytical spindown model for the strongly magnetised and rapidly rotating
PNS to investigate the role of central engine properties such as the surface
dipole field strength, initial rotation period, and jet opening angle on the
interactions and dynamical evolution of the jet-cocoon system. With this model,
we determine the properties of the relativistic jet, the mildly-relativistic
cocoon, and the collimation shock in terms of system parameters such as the
time-dependent jet luminosity, injection angle and density profile of the
stellar medium. We also analyse the criteria for a successful jet breakout, the
maximum energy that can be deposited into the cocoon by the relativistic jet,
and structural stability of the magnetised outflow relative to local
instabilities. Lastly, we compute the high-energy neutrino emission as these
magnetised outflows burrow through their progenitors. Precursor neutrinos from
successful GRB jets are unlikely to be detected by IceCube, which is consistent
with the results of previous works. On the other hand, we find high-energy
neutrinos may be produced for extended progenitors like blue and red
supergiants, and we estimate the detectability of neutrinos with
next-generation detectors such as IceCube-Gen2.
