There's really no relation to cold summers and cold winters. It has no effect whatsoever. This winter could end up being the mildest in the past 100 years, or maybe the coldest ever, who knows, but the current summer really has no part to play.
Not necessarily true. Stations in the south east like Benson recorded -18C temperatures last winter and regularly feature as one of the coldest in the UK, especially in months such as March. Yeovilton in the South West also recorded -17C in the winter of 2009 - 2010. Not sure why these places get so cold yet so low lying and so far south. Even a place in Buckinghamshire near London recorded -19c last December, of course daytime temperatures are much higher and snow doesn't last long compared to here.