I think its normal for the phone to get warm during and just after an update. Once the operating system update completes, then the temperature should gradually return to normal.
As for battery draining fast, there are many usage patterns and apps that could be contributing to that. Screen on time, and cell signal quality tend to be the two most common factors that can contribute to fast battery drain.
Settings/Battery will provide a good view for what usage pattern or apps are consuming battery. I also use GSam Battery Monitor Pro as it gives even more details about how the battery is impacted by phone and app usage patterns and other factors.