We study deformation and defects in thin, flexible sheets with crystalline order using a coarse-grained Phase-Field Crystal (PFC) model. The PFC model describes crystals at diffusive timescales through a continuous periodic field representing the atomic number density. In its amplitude expansion (APFC), a coarse-grained description featuring slowly varying fields retaining lattice deformation, elasticity, and an advanced description of dislocations is achieved. We introduce surface PFC and APFC models in a convenient height formulation encoding normal deformation. With this framework, we then study general aspects of the buckling of strained sheets, defect nucleation on a prescribed deformed surface, and out-of-plane relaxation near dislocations. We benchmark and discuss our results by looking at the continuum limit for buckling under elastic deformation, and at evidence from microscopic models for deformation at defects and defect arrangements. We shed light on the fundamental interplay between lattice distortion at dislocations and out-of-plane deformation by looking at the effect of the annihilation of dislocation dipoles. The scale-bridging capabilities of the devised mesoscale framework are also showcased with the simulation of a representative thin sheet hosting many dislocations.