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).
