In this paper, a new methodology for choosing design parameters of level-crossing analog-to-digital converters (LC-ADCs) is presented that improves sampling accuracy and reduces the data stream rate. Using the MIT-BIH Arrhythmia dataset, several LC-ADC models are designed, simulated and then evaluated in terms of compression and signal-to-distortion ratio. A new one-dimensional convolutional neural network (1D-CNN) based classifier is presented. The 1D-CNN is used to evaluate the event-driven data from several LC-ADC models. With uniformly sampled data, the 1D-CNN has 99.49%, 92.4% and 94.78% overall accuracy, sensitivity and specificity, respectively. In comparison, a 7-bit LC-ADC with 2385Hz clock frequency and 6-bit clock resolution offers 99.2%, 89.98% and 91.64% overall accuracy, sensitivity and specificity, respectively. It also offers 3x data compression while maintaining a signal-to-distortion ratio of 21.19dB. Furthermore, it only requires 49% floating-point operations per second (FLOPS) for cardiac arrhythmia classification in comparison with the uniformly sampled ADC. Finally, an open-source event-driven arrhythmia database is presented.