What is a source of worldwide temperature data that also includes extreme weather?
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.