The decarbonisation of residential heat through integration with the power system and deployment of refurbishment policies is at the core of European energy policies. Yet, heat-electricity integration may be challenged, in practice, by the large variability of heat demand across weather years. Current approaches for residential heat demand simulation fail to provide insights about the extent of such variability across many weather years and about the benefits potentially brought about by nearly zero-energy buildings. To fill this gap, this work develops an open-source space-heating demand simulation workflow that is applicable to any country's building stock. The workflow, based on a well-established lumped-parameter thermodynamic model, allows capturing sub-national weather-year variability and the mitigation effects of refurbishment. For Italy, different weather years lead to variations in heat demand up to 2 TWh/day, lasting for several days. Moreover, some weather regimes produce spatial asymmetries that may further complicate heat-electricity integration. The refurbishment of about 55% of buildings constructed before 1975 could substantially mitigate such oscillations, leading to a 31-37% reduction of yearly heat demand, primarily in colder regions. Intra-day heat demand variations, driven by user behaviour, are not substantially impacted by refurbishment, calling for the simultaneous deployment of flexible heat generating technologies.