This paper simultaneously addresses three limitations associated with conventional skeleton-based action recognition; skeleton detection and tracking errors, poor variety of the targeted actions, as well as person-wise and frame-wise action recognition. A point cloud deep-learning paradigm is introduced to the action recognition, and a unified framework along with a novel deep neural network architecture called Structured Keypoint Pooling is proposed. The proposed method sparsely aggregates keypoint features in a cascaded manner based on prior knowledge of the data structure (which is inherent in skeletons), such as the instances and frames to which each keypoint belongs, and achieves robustness against input errors. Its less constrained and tracking-free architecture enables time-series keypoints consisting of human skeletons and nonhuman object contours to be efficiently treated as an input 3D point cloud and extends the variety of the targeted action. Furthermore, we propose a Pooling-Switching Trick inspired by Structured Keypoint Pooling. This trick switches the pooling kernels between the training and inference phases to detect person-wise and frame-wise actions in a weakly supervised manner using only video-level action labels. This trick enables our training scheme to naturally introduce novel data augmentation, which mixes multiple point clouds extracted from different videos. In the experiments, we comprehensively verify the effectiveness of the proposed method against the limitations, and the method outperforms state-of-the-art skeleton-based action recognition and spatio-temporal action localization methods.