Average winter temperatures could well go down and average summer temperatures could go up. There is no hard coupling between the two with some fixed delta, rather that extremes can get more extreme.

This is, in my humble opinion, the most important aspect to climate change.

Our entire civilization is built on the assumption/requirement that there are enough areas of land that will remain within certain atmospheric parameters for enough contiguous months to grow a lot of food. Technology has widened the atmospheric parameters a bit, and has increased the yield enormously.

Climate change is the result of more energy in the biosphere. This energy is in the process of increasing the (effective) random variabilities of all atmospheric parameters, globally.

Our atmosphere has always been a chaotic system, and when the necessary parameters weren't met for a given region for enough time, then a lot of people migrated and starved.

Adding more energy into a chaotic system tends to make it even more chaotic.

As you noted, these permutations go in both directions.

More unusual heat waves.

More unusual cold waves.

More unusual droughts.

More unusual rain/flooding.

More unusual storms, both warm and cold.

And all of these things globally.

I think there's non-trivial chance that climate change will, on average, cut world-wide food production by a substantial amount over the next two decades. If that is indeed the case, it will be a lot of downs and ups, just as we're seeing now, but, slowly, more and more severe.

You'd need extreme temperatures for both, though. Remember that much of Europe is close to the sea, that buffers both in winter and in summer. Continental climate doesn't really begin until you hit Eastern Europe.

For the last few years, we've had pretty mild winters. This last one, I didn't use my radiators at all because the house is very well insulated and the temperatures didn't drop that low.

Well but that would stand in conflict with the greenhouse-effect, or does the C02 go away in winters?

An easier way to understand this whole climate debate as 'a rise in temperature means there is more energy available in the atmosphere'. That energy can have many effects. It can cause local heating, but it can also move large airmasses around and if those airmasses are colder than the normal for that region the movement will cause a huge swing in temperature.

This is one of the main reasons the average 2.5 degree celsius change (which on an absolute scale would not make a huge difference in the first place, most people would hardly notice) is such a huge affair.

It can cause local variations that are a very large multiple of the 2.5 degrees, much like a heatpump does not require a whole lot of energy to move a much larger amount of energy around.

The greenhouse effect requires sunlight to work. In the winters, the poles do not get any sunlight. The cold air from the poles will travel to make other places colder.

The greenhouse effect does not require sunlight to work. What matters is how much of the energy radiated from the surface of the earth reaches space and how much is reflected back down. https://en.wikipedia.org/wiki/Greenhouse_effect

This is true, but what with sunlight being by far the largest factor in energy being radiated back from the surface of the earth you can safely conclude that in practice the greenhouse effect is to a large extent resulting in solar incident radiation being trapped. In other words: without sunlight nobody would be worrying about the greenhouse effect. In fact, without sunshine we'd love the greenhouse effect, reducing the albedo of the planet as a whole would be the only way to survive as a species (likely we'd fail).