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We use hybrid density functional calculations to assess n-type doping in monoclinic (Al$_x$Ga$_{1-x}$)$_2$O$_3$ alloys. focus on Si, the most promising donor dopant, and study structural properties, formation energies charge-state transition levels of its various configurations. also explore impact C H, which are common impurities metal-organic chemical vapor deposition (MOCVD). In Ga$_2$O$_3$, Si$_{Ga}$ is an effective shallow donor, but Al$_2O_3$ Si$_{Al}$ acts as a DX center with (+/-) level band gap. Interstitial H behaves compensating acceptor Al$_2O_3$. Interpolation indicates that Si up 70% Al, it can be compensated by already at 1% Al. diffusivity complex formation. Si$_{cation}$-H complexes have relatively low binding energies. Substitutional cation site stable negative charge state when x>5%. O (C$_O$) always (Al$_x$Ga$_{1-x}$)$_2$O$_3$, will incorporate only under O-poor conditions. C$_O$-H actually more easily, explaining observations C-related compensation Ga$_2$O$_3$ grown MOCVD. investigate C$_{cation}$-H complexes, finding they high act acceptors x>56%; otherwise just passivates unintentional donors. C-H explains why MOCVD exhibit record-low free-carrier concentrations, spite unavoidable incorporation C. Our highlights that, while suitable ALGO alloys, control essential avoid compensation.