  To guide improved properties coincident with reduction of critical materials
in permanent magnets, we investigate via density functional theory (DFT) the
intrinsic magnetic properties of a promising system,
$R$(Fe$_{1-x}$Co$_{x}$)$_{11}$Ti$Z$ with $R$=Y, Ce and interstitial doping
($Z$=H, C, N). The magnetization $M$, Curie temperature $T_\text{C}$, and
magnetocrystalline anisotropy energy $K$ calculated in local density
approximation to DFT agree well with measurements. Site-resolved contributions
to $K$ reveal that all three Fe sublattices promote uniaxial anisotropy in
YFe$_{11}$Ti, while competing anisotropy contributions exist in YCo$_{11}$Ti.
As observed in experiments on $R$(Fe$_{1-x}$Co$_{x}$)$_{11}$Ti, we find a
complex nonmonotonic dependence of $K$ on Co content, and show that anisotropy
variations are a collective effect of MAE contributions from all sites and
cannot be solely explained by preferential site occupancy. With interstitial
doping, calculated $T_\text{C}$ enhancements are in the sequence of N$>$C$>$H,
with volume and chemical effects contributing to the enhancement. The uniaxial
anisotropy of $R$(Fe$_{1-x}$Co$_{x}$)$_{11}$Ti$Z$ generally decreases with C
and N; although, for $R$=Ce, C doping is found to greatly enhance it for a
small range of 0.7$<$$x$$<$0.9.
