The continuous flow of gas and dark matter across scales in the cosmic web can generate correlated dynamical properties of haloes and filaments (and the magnetic fields they contain). With this work, we study the halo spin properties and orientation with respect to filaments, and the morphology of the magnetic field around these objects, for haloes with masses in the range ∼108 − 1014 M⊙ and filaments up to ∼8 Mpc long. Furthermore, we study how these properties vary in presence, or lack thereof, of different (astro)physical processes and with different magnetic initial conditions. We perform cosmological magnetohydrodynamical simulations with the Eulerian code Enzo and we develop a simple and robust algorithm to study the filamentary connectivity of haloes in three dimensions. We investigate the morphological and magnetic properties and focus on the alignment of the magnetic field along filaments: our analysis suggests that the degree of this alignment is partially dependent on the physical processes involved, as well as on magnetic initial conditions. We discuss the contribution of this effect on a potential attempt to detect the magnetic field surrounding these objects: we find that it introduces a bias in the estimation of the magnetic field from Faraday rotation measure techniques. Specifically, given the strong tendency we find for extragalactic magnetic fields to align with the filaments axis, the value of the magnetic field can be underestimated by a factor ∼3, because this effect contributes to making the line-of-sight magnetic field (for filaments in the plane of the sky) much smaller than the total one.