Ruthenium dioxide (RuO$_2$) has been proposed as a prototypical metallic $d$-wave altermagnet, a Néel-ordered compensated antiferromagnetic state exhibiting nonrelativistic momentum-dependent spin splitting; yet, its magnetic ground state remains controversial both theoretically and experimentally. Using comprehensive first-principles calculations, we investigate RuO$_2$ thin films with (110), (100), and (001) orientations, both (un)strained freestanding and supported on a TiO$_2$ substrate. We show that emergent magnetic moments in RuO$_2$ thin films are highly fragile, strongly influenced by strain, surface orientation, and atomic relaxation, while also being highly sensitive to the choice of the Brillouin-zone integration scheme. We find that none of the thin film structures considered can stabilize a compensated antiferromagnetic order; therefore, an altermagnetic ground state cannot be realized. Instead, substrate-supported RuO$_2$ films on TiO$_2$ exhibit pronounced layer- and site-dependent magnetic moment variations and incomplete compensation between the two antiferromagnetically coupled Ru moments, yielding a \emph{ferrimagnetic-like} behavior. On the other hand, freestanding RuO$_2$ films display complex magnetic structures depending on their orientation and applied strain, with distinct behavior at the surfaces and in the inner layers. Our results reconcile conflicting theoretical and experimental reports and underscore the sensitivity of RuO$_2$ magnetism to structural and methodological details.