Shell structure and spontaneous decay of $Z=124$ isotopes are systematically investigated using the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) with the PC-PK1 density functional. By accounting for deformation and continuum effects, the neutron drip line is predicted to be located at $N=320$, and eight potential multi-neutron emitters beyond the neutron drip line are identified. The results, compared with the spherically constrained case, highlight the critical role of deformation effects in determining the ground-state properties of superheavy nuclei. In addition, potential neutron shell closures emerge at $N=258$ and $350$, alongside subshell closures at $N=232$ and $320$. The results disfavor $Z=124$ as a possible proton magic number and suggest $Z=120$ and $138$ as more favorable candidates for proton shell closures in the superheavy region. Finally, the competition between $\alpha$ decay, $\beta$ decay, and spontaneous fission is analyzed by calculating half-lives within various semi-empirical formulas. The results indicate that $\alpha$ decay is the predominant mode on the proton-rich side, whereas spontaneous fission and ${\beta }^{-}$ decay gradually become dominant with increasing neutron number.