What is a source of worldwide temperature data that also includes extreme weather? I often need to calculate heat balances for large biogas plants, an important factor is outside temperature. I normally use the weather data from Klimadiagramme.de where I can find the mean monthly temperature and precipitation. This allows me to calculate the heat demand of my plant for a given month reasonably well. However, I also need to know extreme weather. I'm not sure what the metric here is, but I'm not looking for "the coldest night in the last 100 years" but "this was the coldest average over 2-3 days in the last x years" or something similiar. 

So, what is a good, free source of climate data that gives me information about extreme weather events as described?
CLIMDEX, the "Datasets for Indices of Climate Extremes" is probably the closest you are going to get, for any pre-computed indices. The 27 indices it includes are listed on the indices page of that site.

Unfortunately, it doesn't include anything like "coldest 3-day average per year", let alone "coldest 3-day average over the last 5 years". I doubt there would be any pre-built dataset with exactly that index, and there is unlikely to be online tool that would calculate unusual indices, because the data sets are huge (HadCRUt4 is 20MB for monthly data over the last 150 years), and unusual indices would have to be calculated on the fly - something that's just not feasible on the 'web.

I would guess that you'd have to calculate that yourself, from a raw daily temperature dataset. Unfortunately, I don't think such a thing exists on a global scale. However, a reanalysis product like the NCEP/NCAR Reanalysis is probably good enough for your purposes. However, you'll need to be familiar with large data analysis - the daily surface temperature dataset is over 450MB. 

Depending on your exact usage requirements, you might be better off just looking up some of the historical records for neighbouring meteorological stations and using those.