We study the superfluid weight $D^s$ and Berezinskii-Kosterlitz-Thouless (BKT) transition temperatures $T_{BKT}$ in case of exotic Fulde-Ferrell (FF) superfluid states in lattice systems. We consider spin-imbalanced systems with and without spin-orbit coupling (SOC) accompanied with in-plane Zeeman field... By applying mean-field theory, we derive general equations for $D^s$ and $T_{BKT}$ in the presence of SOC and the Zeeman fields for 2D Fermi-Hubbard lattice models, and apply our results to a 2D square lattice. We show that conventional spin-imbalanced FF states without SOC can be observed at finite temperatures and that FF phases are further stabilized against thermal fluctuations by introducing SOC. We also propose how topologically non-trivial SOC-induced FF phases could be identified experimentally by studying the total density profiles. Furthermore, the relative behavior of transverse and longitudinal superfluid weight components and the role of the geometric superfluid contribution are discussed. (read more)