Real-time quantification of residential building energy flexibility is needed to enable a cost-efficient operation of active distribution grids. A promising means is to use the socalled flexibility envelope concept to represent the time-dependent and inter-temporally coupled flexibility potential. However, existing optimization-based quantification entails high computational burdens limiting flexibility utilization in real-time applications, and a more computationally efficient approach is desired. Also, the communication of a flexibility envelope to system operators in its original form is data-intensive. To address these issues, this paper first trains several machine learning models based on historical quantification results for online use. Subsequently, probability distribution functions are proposed to approximate the flexibility envelopes with significantly fewer parameters, which can be communicated to system operators instead of the original flexibility envelope. The results show that the most promising prediction and approximation approaches allow for a minimum reduction of the computational burden by a factor of 9 and of the communication load by a factor of 6.6, respectively.