Flexoelectric effect, or flexocoupling, is a basic property of virtually all solid-state materials. Direct flexoeffect couples polarization and mechanical strain via their gradients, effectively describing a poling of the bulk material when it is subjected to the inhomogeneous stress, an inverse effect exists too. The effect itself is weak and only becomes remarkable in thin films and nanostructures, however, it may influence their macroscopic properties. In this work, the effect of flexoelectricity on domain structures in thin ferroelectric films is studied using Landau-Ginzburg-Devonshire theory. Properties of the structures, including phase transitions, domain and domain wall shapes, domain sizes and periodicity, are found to be impacted to a different extent by the flexocoupling. Surface screening is another film property that affects domain structures. The screening can be caused by metallic electrodes, as well as by charges adsorbed from the ambience on the free surface of the film. Modification of the surface charge results in adjustment of domain shapes and the structure type, forcing transitions between stripe domains, closure domains and the single-domain state. This work investigates competition between the two above mentioned mechanisms always present in real films. At larger surface charge compensation, classical stripe domains are formed, while at smaller surface charge compensation, closure domain structures appear which effectively suppress bound polarization charges at the free film surface. Flexoeffect leads to formation of enhanced stress regions at the clamped bottom surface of the film and to appearance of metastable nanodomains in the bulk of films of higher thicknesses.