  In this paper, we perform a systematic {\it ab initio} study of two principal
spin-related phenomena, namely, anomalous Hall effect and current spin
polarization, in Co$_2$Fe-based Heusler compounds Co$_2$FeX (X = Al, Ga, In,
Si, Ge, Sn) within the generalized gradient approximation (GGA). The accurate
full-potential linearized augmented plane-wave method is used. We find that the
spin-polarization of the longitudinal current ($P^L$) in Co$_2$FeX (X = Al, Ga,
In, Al$_{0.5}$Si$_{0.5}$ and Sn) is $\sim$100 \% even though that of the
electronic states at the Fermi level ($P^D$) is not. Further, the other
compounds also have a high current spin polarization with $P^L > 85$ \%. This
indicates that all the Co$_2$FeX compounds considered are promising for
spin-transport devices. Interestingly, $P^D$ is negative in Co$_2$FeX (X = Si,
Ge and Sn), differing in sign from the $P^L$ as well as that from the transport
experiments. Secondly, the calculated anomalous Hall conductivities (AHCs) are
moderate, being within 200 S/cm, and agree well with the available experiments
on highly L2$_1$ ordered Co$_2$FeSi specimen although they differ significantly
from the reported experiments on other compounds where the B2 antisite
disorders were present. Surprisingly, the AHC in Co$_2$FeSi decreases and then
changes sign when Si is replaced by Ge and finally by Sn. Third, the calculated
total magnetic moments agree well with the corresponding experimental ones in
all the studied compounds except Co$_2$FeSi where a difference of 0.3
$\mu_B$/f.u. exists. We also perform the GGA plus on-site Coulomb interaction
$U$ calculations in the GGA+$U$ scheme. We find that including the $U$ affects
the calculated total magnetic moment, spin polarization and AHC significantly,
and in most cases, results in a disagreement with the available experimental
results.
