Understanding extreme climate events and their probability is key for the study of climate change impacts, risk assessment, adaptation, the protection of living beings, and renewable electricity production. Extreme heatwaves are, and likely will be in the future, among the deadliest weather events. We study extreme heat waves and extremes of low renewable electricity production over different areas of the mid-latitude Earth atmosphere. We show that their occurrences are associated to hemispheric scale patterns, most of the time dominated by wavenumber three dynamically quasi-stationary patterns of turbulent Rossby waves, that depend on the extreme event location. We stress the relevance of these patterns for recently observed extreme heat waves and their predictability potential. For short return time heatwaves, those extreme teleconnection patterns and their dynamics are robustly predicted by a hierarchy of models and are consistent with reanalysis datasets. For longer return times, the reanalysis datasets are too short to give statistically significant patterns. A timeseries analysis shows that extreme event indices defined at synoptic spatial and temporal scales are often rather well described by Gaussian stochastic processes. Those Gaussian processes can reproduce well return time plots even for very rare events.