Massive black hole (BH) mergers will be key targets of future gravitational wave and electromagnetic observational facilities. In order to constrain BH evolution with the information extracted from BH mergers, one must take into account the complex relationship between the population of merging BHs and the global BH population. We analysed the high-resolution cosmological radiation-hydrodynamics simulation OBELISK, run to redshift z=3.5, to study the properties of the merging BH population, and its differences with the underlying global BH population in terms of BH and galaxy properties. In post-processing, we calculated dynamical delays between the merger in the simulation at the resolution limit and the actual coalescence well below the resolution scale. We find that merging BHs are hosted in relatively massive galaxies with stellar mass M*>~10^9^M_{sun}_. Given that galaxy mass is correlated with other BH and galaxy properties, BH mergers tend to also have a higher total BH mass and higher BH accretion rates than the global population of main BHs. These differences generally disappear if the merger population is compared with a BH population sampled with the same galaxy mass distribution as merger hosts. Galaxy mergers can temporarily boost the BH accretion rate and the host's star formation rate, which can remain active at the BH merger if sub-resolution delays are not taken into account. When dynamical delays are taken into account, the burst has generally faded by the time the BHs merge. BH spins are followed self-consistently in the simulation under the effect of accretion and BH mergers. We find that merging BHs have higher spins than the global population, but similar or somewhat lower spins compared to a mass-matched sample. For our sample, mergers tend to decrease the spin of the final BH remnant. Massive black-hole (BH) mergers are predicted to be powerful sources of low-frequency gravitational waves (GWs). Coupling the detection of GWs with an electromagnetic (EM) detection can provide key information about merging BHs and their environments as well as cosmology. We study the high-resolution cosmological radiation-hydrodynamics simulation OBELISK, run to redshift z=3.5, to assess the GW and EM detectability of high-redshift BH mergers, modelling spectral energy distribution and obscuration. For EM detectability, we further consider sub-grid dynamical delays in postprocessing. We find that most of the merger events can be detected by LISA, except for high-mass mergers with very unequal mass ratios. Intrinsic binary parameters are accurately measured, but the sky localisation is poor generally. Only ~40% of these high-redshift sources have a sky localisation better than 10deg^2^. Merging BHs are hard to detect in the restframe UV since they are fainter than the host galaxies, which at high redshift are star-forming. A significant fraction, 15-35%, of BH mergers instead outshine the galaxy in X-rays, and about 5-15% are sufficiently bright to be detected with sensitive X-ray instruments. If mergers induce an Eddington-limited brightening, up to 30% of sources can become observable. The transient flux change originating from such a brightening is often large, allowing 4-20% of mergers to be detected as EM counterparts. A fraction, 1-30%, of mergers are also detectable at radio frequencies. Transients are found to be weaker for radio-observable mergers. Observable merging BHs tend to have higher accretion rates and masses and are overmassive at a fixed galaxy mass with respect to the full population. Most EM-observable mergers can also be GW-detected with LISA, but their sky localisation is generally poorer. This has to be considered when using EM counterparts to obtain information about the properties of merging BHs and their environment.