We present a comprehensive study of the anisotropic magnetoresistance (AMR) in single-crystal $\mathrm{F}{\mathrm{e}}_{1-x}\mathrm{N}{\mathrm{i}}_x\left(001\right)$ alloy films with body-centered-cubic (bcc) structures, revealing a giant current-direction dependence and significant temperature-dependent behavior. In $\mathrm{F}{\mathrm{e}}_{0.8}\mathrm{N}{\mathrm{i}}_{0.2}$, the ratio between the AMR values for current along the [100] and [110] directions reaches 87 at 5 K. This ratio exhibits a strong temperature dependence, increasing nearly an order of magnitude as the temperature decreases from 300 K to 5 K. Additionally, we identify ultralow Gilbert damping ($\alpha \sim 2.3\times {10}^{-3}$) with negligible dependence on the magnetization orientation. The intrinsic anomalous Hall conductivity is found to decrease monotonically with Ni concentration. These results not only advance our understanding of FeNi alloys but also provide valuable insights for developing materials with customized magnetoresistance and low damping for magnetic sensing and data storage applications.

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