Right, so this is in most respects a better question for /r/askanthropology than it is for /r/askhistory, but there are some historical aspects to it that can be answered. Keep in mind that the following is nowhere near comprehensive; it is only a small portion of a possible answer.

First of all, it is untrue that developed countries tend to be "cold-weather". Developed countries today tend to be in temperate climates, with, yes, cold winters, but also reasonably long growing seasons. Save for Iceland, no highly developed country exists that lacks this; the Nordic countries' populations are concentrated in southern regions that have climatic conditions similar to their continental neighbors. Truly cold regions, such as northern Canada, Russia, and Scandinavia, are either undeveloped or are developed as a consequence of government subsidy or valuable resource extraction. Leaving that aside though, what unique disadvantages do warmer regions have?

One of the biggest is that a subset of these, tropical regions, have historically had low population densities. Urbanization and ease of transfer (for goods, ideas, people) across territory makes development easier; for example, the entirety of South America had a smaller population than the United States from the early 1800s until relatively recently, while the population of all of Africa from 1800 to the mid 20th century struggled to keep pace with just Russia.

The first and foremost cause of this low density is disease, particularly those caused by internal parasites (tapeworms, roundworms, etc.) or external parasites (mosquitos, ticks, etc.). Freezing winters either kill or mitigate the effects of most parasites, which do not only spread deadly diseases like malaria (which has killed, by far, the most people in history of any disease), but also can cause delays/damage to brain development and physical strength. Malaria is an interesting case because of the development of resistance to malaria among many people of West African descent, as this resistance comes as the cost of a high risk of sickle-cell disease. Tropical diseases, while most damaging in the tropics, have caused tremendous suffering in sub-tropical regions like the Middle East and Southern Europe. Malaria was a huge killer in Rome, for example, prior to the draining of the Pontine marshes in the 1920s-30s. Many of these diseases were native to the Old World; the transfer of diseases such as Yellow Fever, Dengue, and Malaria reduced the habitability of much of South and Central America compared to the period prior to European contact.

An issue specific to the tropics is poor soil quality, which also has contributed to low population density. Tropical soils cannot sustain large-scale agriculture without intensive use of fertilizers. Prior to the large scale development and exploitation of both natural and artificial fertilizers, high density regions in tropical regions tended to be located in volcanic areas, such as Java and the Eastern Congo/Rwandan regions.

A semi-related issue comes when considering climatic shifts over the past couple thousand years, since the (Old World) civilizations you mention. The Middle East and North Africa have become significantly drier since the Roman period. This did not make sophisticated societal development in those regions impossible, but it did mean that the people and polities in those regions were working with less optimal conditions than their Carthaginian, Roman, etc. forbears were.

None of this explains the greater question you have asked, and it ignores extremely important factors such as imperialism writ large, differences between different styles of colonial rule, specific historical events/phenomena/personas, and all sorts of other factors that have contributed to the modern day condition of high development in temperate regions and lower development in warmer regions. What it does do is give some underlying factors, some of which have remained static, and some of which have changed since the ancient time period.

Some population stats: http://www.thuto.org/ubh/ub/h202/wpop1.htm

Discussion of the changing climate of N. Africa: http://www.geo.hunter.cuny.edu/courses/geog347/articles/decline_human.pdf (He ultimately concludes that human activity via hunting/soil erosion might be able to explain the decline of N. Africa without needing a climatic explanation, but if that is the case it remains true that N. Africa is less conducive to "higher development" currently, just for a different reason)

A more pro-climate take: https://surface.syr.edu/cgi/viewcontent.cgi?referer=https://en.wikipedia.org/&httpsredir=1&article=1017&context=suscholar

There is a pretty clear trend of countries with cold weather countries being more developed than those with pleasant climates.

That is not really true. If we are measuring by Human Development Index, it becomes pretty clear that development levels do not map neatly to latitude or climate. If we look at 2018 Human Development Index map we don't see a strong sorting by latitude. Countries in South America score consistently higher on HDI scores than African countries despite having comparable climates and being at similar latitudes.

Ditto, within Africa you have countries like Gabon which have much better HDI scores than neighboring Democratic republic of Congo, despite having similar climates. And contra to your statement that "North and South Africa are more developed than Equatorial africa", by these metrics, Equatorial african countries like Republic of Congo, Gabon, Cameroon, Ghana are more developed than southern african countries like Madagascar, Malawi, Mozambique and Zimbabwe.

Or, look at Malaysia which sits on the equator and is in the "very high development" category.

Clearly there are exceptions, but the trend holds.

In my opinion, there are too many exceptions to say your trend is valid.

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And of course, you rightly point out that in past eras there existed highly complex societies in regions which are now in countries with medium or low HDI scores.

So, no, it was not "natural selection" or geographical determinism that led to the current world order.

As to why things are as they are, it's a complex question. Numerous human decisions, and broad social and historical trends all came together to lead to some countries becoming very rich and having high levels of development, and other countries being poor, unstable, and having bad infrastructure.

A major piece of any answer must contend with the reality that Europeans colonized virtually the whole world, and that historical moment (from perhaps 1500 AD to 1950 AD) was when the change happened from ancient warm-weather civilizations to the current economic world order.

Perhaps it is unsurprising that this question of "why did europeans start to take over the world" has been asked before, and /u/Profrhodes gives a good overview of the schools of thought here.

I find /u/Profrhodes synopsis of Eric Hobsbawm that:

Much better historians like Eric Hobsbawm (The Age of Revolution: Europe 1789–1848) provide much more solid and rounded explanations. As he states, it is better to argue that the developments during the three centuries following 1492, including the creation and expansion of mercantile empires, the creation of a captive periphery in the Americas, and the proto-capitalist economies of the Carribean and Americas (in the form of plantation systems where whole sections of the society were geared towards production) all provided the basis for the Industrial Revolution to bring Europe to the fore. The New World in particular eased the land constraints within Europe (in the shape of 'ghost area' colonies) and allowed Britain in particular to access coal and thereby shift from the biological/organic ancien regime to the inorganic industrial economy. At the same time, from 1815 onwards European colonial empires saw huge swathes of land being colonized by European industrial powers, and contributing to the economic expansion. From that point onwards (1800) European hegemony was guaranteed.

very intriguing. The implication of Hobsbawm's argument here is that the economic gains that resulted from conquest of the Americas laid the basis for the first Industrial Revolution and hence European colonization of the rest of the world.

There is an entire genre of literature that contends that a great portion of African underdevelopment in the 21st century occurs because of 19th and early 20th century colonial economic exploitation. Probably the most famous work in that genre is Walter Rodney's book How Europe underdeveloped Africa.

Now, there is of course more to be said. There is individual and continental difference among post-colonial countries and their economic and developmental trajectories. South America gained independence in the 1820s and so has had almost 200 years of post-colonial economic development. Most Asian countries gained independence in the 1940s or 1950s, and since the 1990s have had great development gains. On the other hand, most African countries gained independence in the 1960s (or later for some southern African countries), and enjoyed strong economic growth and development into the 1980s, before seeing a "lost generation of growth" into the 2000s.

So, clearly post-colonial leadership and economic policy have mattered. Some countries like Vietnam, Thailand, and Malaysia have been able to attract foreign investment and seen strong economic growth and development gains in the past 20 years. At the same time, certain countries like Zimbabwe or Iraq or Syria had very healthy economies and fairly high levels of development, but have slid backwards in the past 20 years. Still other countries like Afghanistan or DRC or Myanmar have had persistent conflict for long periods, cutting off opportunities for economic growth and human development.