The field of spintronics is emerging at blistering pace due to its promising room-temperature applications in low-power logic-based devices. In this work, we investigate the suitability of ferromagnetic (FM) La0.7Sr0.3MnO3 (LSMO) thin film/ferroelectric (FE) Pb(Mg0.33Nb0.67)O3-PbTiO3 (PMN-PT) heterostructures for spintronic device applications through characterizing the static and dynamic magnetic properties. Due to the magnetoelastic coupling at the interface, LSMO/PMN-PT(001) is found to exhibit a 4-fold symmetry of magnetic anisotropy, while LSMO/PMN-PT(111) shows isotropic behavior at room temperature. The direction dependent Gilbert damping constant (α) estimated for the LSMO/PMN-PT(001) sample clearly shows that its values are smaller along the magnetically hard axis than along the easy axis, while no significant variation is observed for the LSMO/PMN-PT(111) sample. The electric field induced variation of α also demonstrate that the Gilbert damping is closely related to the magnetic anisotropy. The present results clearly indicate that it will be possible to realize electric field controlled spin dynamics in FM/FEs to develop futuristic spintronic devices.