Theoretically and experimentally, MXenes consisting of Mo and C have aroused much interest due to superconductivity in their films and even monolayer forms. Here, based on first-principles calculations, we systematically calculate the electronic structure, phonon dispersion, and electron-phonon coupling (EPC) of monolayer Mo2C (both T- and H-phases), Mo3C2, and Mo3C3. The results show that H-MoxCy (x = 2 or 3, y = 1-3) always have lower total energies than their corresponding T phase and other configurations. All these two-dimensional (2D) molybdenum carbides are metals and some of them display weak phonon-mediated superconductivity at different superconducting transition temperatures (Tc). The Mo 4d-orbitals play a critical role in their electronic properties and the Mo atomic vibrations play a dominant role in their low-frequency phonons, EPC, and superconductivity. By comparison, we find that increasing the Mo content can enhance the EPC and Tc. Besides, we further explore the impact of strain engineering on their superconducting related physical quantities. With increasing biaxial stretching, the phonon dispersions are gradually softened to form some soft modes, which can trigger some peaks of α2F(ω) in the low-frequency region and evidently increase the EPC λ. The Tc of H-Mo2C can be increased up to 11.79 K. Upon further biaxial stretching, charge density waves may appear in T-Mo2C, H-Mo3C2, and H-Mo3C3.