Optical microscopy is an essential tool in biology and medicine. Imaging thin, yet non-flat objects in a single shot (without relying on more sophisticated sectioning setups) remains challenging as the shallow depth of field that comes with high-resolution microscopes leads to unsharp image regions and makes depth localization and quantitative image interpretation difficult. Here, we present a method that improves the resolution of light microscopy images of such objects by locally estimating image distortion while jointly estimating object distance to the focal plane. Specifically, we estimate the parameters of a spatially-variant Point Spread Function (PSF) model using a Convolutional Neural Network (CNN), which does not require instrument- or object-specific calibration. Our method recovers PSF parameters from the image itself with up to a squared Pearson correlation coefficient of 0.99 in ideal conditions, while remaining robust to object rotation, illumination variations, or photon noise. When the recovered PSFs are used with a spatially-variant and regularized Richardson-Lucy (RL) deconvolution algorithm, we observed up to 2.1 dB better Signal-to-Noise Ratio (SNR) compared to other Blind Deconvolution (BD) techniques. Following microscope-specific calibration, we further demonstrate that the recovered PSF model parameters permit estimating surface depth with a precision of $\mathrm {2 \mu \text {m} }$ and over an extended range when using engineered PSFs. Our method opens up multiple possibilities for enhancing images of non-flat objects with minimal need for a priori knowledge about the optical setup.