A robust multichannel speaker diarization and separation system is proposed by exploiting the spatio-temporal activity of the speakers. The system is realized in a hybrid architecture that combines the array signal processing units and the deep learning units. For speaker diarization, a spatial coherence matrix across time frames is computed based on the whitened relative transfer functions (wRTFs) of the microphone array. This serves as a robust feature for subsequent machine learning without the need for prior knowledge of the array configuration. A computationally efficient Spatial Activity-driven Speaker Diarization network (SASDnet) is constructed to estimate the speaker activity directly from the spatial coherence matrix. For speaker separation, we propose the Global and Local Activity-driven Speaker Extraction network (GLASEnet) to separate speaker signals via speaker-specific global and local spatial activity functions. The local spatial activity functions depend on the coherence between the wRTFs of each time-frequency bin and the target speaker-dominant bins. The global spatial activity functions are computed from the global spatial coherence functions based on frequency-averaged local spatial activity functions. Experimental results have demonstrated superior speaker, diarization, counting, and separation performance achieved by the proposed system with low computational complexity compared to the pre-selected baselines.