Sparse keypoint matching based on distinct 3D feature representations can improve the efficiency and robustness of point cloud registration. Existing learning-based 3D descriptors and keypoint detectors are either independent or loosely coupled, so they cannot fully adapt to each other. In this work, we propose a tightly coupled keypoint detector and descriptor (TCKDD) based on a multi-task fully convolutional network with a probabilistic detection loss. In particular, this self-supervised detection loss fully adapts the keypoint detector to any jointly learned descriptors and benefits the self-supervised learning of descriptors. Extensive experiments on both indoor and outdoor datasets show that our TCKDD achieves state-of-the-art performance in point cloud registration. Furthermore, we design a keypoint detector and descriptors-assisted LiDAR odometry and mapping framework (KDD-LOAM), whose real-time odometry relies on keypoint descriptor matching-based RANSAC. The sparse keypoints are further used for efficient scan-to-map registration and mapping. Experiments on KITTI dataset demonstrate that KDD-LOAM significantly surpasses LOAM and shows competitive performance in odometry.