Channel state information (CSI) at transmitter is crucial for massive MIMO downlink systems to achieve high spectrum and energy efficiency. Existing works have provided deep learning architectures for CSI feedback and recovery at the eNB/gNB by reducing user feedback overhead and improving recovery accuracy. However, existing DL architectures tend to be inflexible and non-scalable as models are often trained according to a preset number of antennas for a given compression ratio. In this work, we develop a flexible and scalable learning framework based on a divide-and-conquer approach (DCA). This new DCA architecture can flexibly accommodate different numbers of 3GPP antenna ports and dynamic levels of feedback compression. Importantly, it also significantly reduces computational complexity and memory size by allowing UEs to feedback segmented downlink CSI. We further propose a multi-rate successive convolution encoder with fewer than 1000 parameters. Test results demonstrate superior performance, good scalability, and low complexity for both indoor and outdoor channels.