The KM3NeT research infrastructure is currently under construction in two locations in the Mediterranean Sea. The KM3NeT/ORCA water-Cherenkov neutrino detector off the French coast will instrument several megatons of seawater with photosensors. Its main objective is the determination of the neutrino mass ordering. For the first time, deep convolutional neural networks are employed to achieve three specific reconstruction and classification tasks that constitute a complete alternative analysis pipeline for KM3NeT/ORCA detector data. The values and uncertainties of the energy, direction, and the interaction vertex of the incident neutrinos are reconstructed. The neutrino interaction type is recognized as shower- or track-like, and the main backgrounds to the detection of atmospheric neutrinos are classified and suppressed. Performance comparisons to other machine-learning methods used and maximum-likelihood reconstruction algorithms previously developed for KM3NeT/ORCA are provided. It is shown that this image recognition technique yields competitive results and performance improvements for large-volume neutrino telescopes.