We perform ab initio calculations of the coupling between electrons and small-momentum polar-optical phonons in monolayer transition-metal dichalcogenides of the 2H type: MoS2, MoSe2, MoTe2, WS2, and WSe2. The polar-optical coupling with longitudinal optical phonons, or Frohlich interaction, is fundamentally affected by the dimensionality of the system. In a plane-wave framework with periodic boundary conditions, the Frohlich interaction is affected by the spurious interaction between the two-dimensional (2D) material and its periodic images. To overcome this difficulty, we perform density functional perturbation theory calculations with a truncated Coulomb interaction in the direction perpendicular to the plane of the 2D material. We show that the two-dimensional Frohlich interaction is much stronger than assumed in previous ab initio studies. We provide analytical models depending on the effective charges and dielectric properties of the materials to interpret our ab initio calculations. Screening is shown to play a fundamental role in the phonon-momentum dependency of the polar-optical coupling, with a crossover between two regimes depending on the dielectric properties of the material relative to its environment. The Frohlich interaction is screened by the dielectric environment in the limit of small phonon momenta and sharply decreases due to stronger screening by the monolayer at finite momenta. The small-momentum regime of the ab initio Frohlich interaction is reproduced by a simple analytical model, for which we provide the necessary parameters. At larger momenta, however, direct ab initio calculations of electron-phonon interactions are necessary to capture band-specific effects. We compute and compare the carrier relaxation times associated with the scattering by both LO and A(1) phonon modes. While both modes are capable of relaxing carriers on time scales under the picosecond at room temperature, their absolute importance and relative importance vary strongly depending on the material, the band, and the substrate.