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• Net-zero emissions are modelled for electricity, heating and cooling in Europe. • System cost decreases thanks to heat savings, but not for climate change scenarios. • Lower heat demand disfavours heat pumps as surplus electricity is used by resistors. • Cheaper heat pumps impact resistive heaters substantially, but not the opposite. To limit the global temperature increase to 1.5 °C, fossil-free energy systems will be required eventually. To understand how such systems can be designed, the current state-of-the-art is to apply techno-economical optimisation modelling with high spatial and temporal resolution. This approach relies on a number of climatic, technical and economic predictions that reach multiple decades into the future. In this paper, we investigate how the design of a fossil-free energy system for Europe is affected by changes in these assumptions. In particular, the synergy among renewable generators, power-to-heat converters, storage units, synthetic gas and transmission manage to deliver an affordable net-zero emissions system. We find that levelised cost of energy decreases due to heat savings, but not for global temperature increases. In both cases, heat pumps become less favourable as surplus electricity is more abundant for heating. Demand-side management through buildings' thermal inertia could shape the heating demand, yet has modest impact on the system configuration. Cost reductions of heat pumps impact resistive heaters substantially, but not the opposite. Cheaper power-to-gas could lower the need for thermal energy storage. [ABSTRACT FROM AUTHOR]