Lederhosen highlight a curious quirk of language. The german word for trouser is hose. Lederhosen means leather pants. Lederhosen seem to be analogous to english breeches.
Around the time breeches succeeded braies is when the transition from hosen (medieval stockings) to trousers began. Hosen were long woven wool socks that were attached by suspenders to a medieval man's belt or underwear, in a similar fashion to modern womens suspended stockings (erotic/novelty legwear). The medieval mans underwear were called braies which turned into early modern breeches. Here is an interesting wikipedia quote about this:
"Before the 1590s, stockings were made of woven cloth. The first knitting machines were for making stockings."
This suggests that the transition from hosen/braies to socks/trousers, whereby braies elongated into breeches and thereupon into trousers, and hosen shortened into socks, took place at around the same time as when the knitting machine was invented.
We know that in the napoleonic times the revolution was spearheaded by sans-culottes, or "without breeches." Culottes refers to the fancy white knee length breeches that we are familiar with from photos of 18th century soldiers like George Washington or redcoats. Below the knee were knee high socks or boots. At the same time the sans culottes disdained this aristocratic style and began to wear ankle length trousers instead.
This was during the industrial revolution. As mentioned the knitting machine was invented 2cent earlier ("to make stockings") and during the 1700s machines were invented for weaving. The clothing industry was being reshaped. A few decades later sewing machines would become prevalent. Additionally colonialism introduced imports of foreign fabric like cotton that had different properties from native european fabrics like wool, linen, or hemp. Some time during the 1800s and 1900s cotton had become so ubiquitous that all legwear, both underwear and trousers were made from cotton.
My quest here is, to examine more fully the development of modern clothing. If anyone can shed light on this transition, I would be grateful. How did we get from braies to breeches to trousers to pants? How did we get from hosen to stockings to socks? What role did industrialization and colonization play in this process?
Also, as a postscript, I will note that, having experimented with making my own wool trousers, I believe one reason why the braies/hosen combo may have been pragmatic, is that you could presumably affix multiple layers of wool hosen to a single pair of braies. This would be very useful in winter time. This is because wearing multiple wool trousers is rather uncomfortable. I am curious as to what did dark age germans (who wore wool trousers) wear in deep winter, and also what did industrial europeans (who, thousands of years later, also wore wool trousers) wear in deep winter. For example, did russians in 1812 put on multiple trousers? If so, did each pair have its own belt? Did they make quilted trousers? This reminds me that concurrently in north america, trappers wore leather leggings, which conceivably could have been made from furs in winter; fur hosen.
To some extent, you're answering your own question, but I think that your lack of familiarity with historic fashion is leading you down the wrong path in some respects - namely, the idea that braies (which were underwear) turned into breeches and hose turned into stockings. This past question, How did medieval pants work? has answers by myself and /u/somecrazynerd that explain how medieval long hose was eventually split into upper and nether hose, the upper hose becoming breeches and the nether hose stockings.
Following that, I also have an answer to I'm a young, well-to-do Englishman of fashion in the 1810s. During which years of the decade am I most likely to wear knee breeches as opposed to long pants, and vice-versa? Here I discuss the gradual phasing out of formal breeches in the 1810s, which answers the second half of the question: elongated breeches - pantaloons - became a staple of informal dress in the early nineteenth century and were eventually considered formal enough to be worn on all occasions.
(You also ask about going from trousers to pants, but these two words are synonyms, so I'm not entirely sure what is being referred to.)
This suggests that the transition from hosen/braies to socks/trousers, whereby braies elongated into breeches and thereupon into trousers, and hosen shortened into socks, took place at around the same time as when the knitting machine was invented.
I am not quite sure why you are coming to this conclusion. As noted in the previous answer, the split actually preceded the invention of the knitting frame by more than a century. The knitting frame's invention itself reflected the fact that women were already hand-knitting stockings as a cottage industry (the patent was denied by Queen Elizabeth so as not to deprive them of their jobs), but at the same time, stockings made of cut and pieced wool, linen, and silk were also being worn. By cutting pieces on the bias of the fabric, they were able to be reasonably stretchy and conform to the shape of the leg.
We know that in the napoleonic times the revolution was spearheaded by sans-culottes, or "without breeches." Culottes refers to the fancy white knee length breeches that we are familiar with from photos of 18th century soldiers like George Washington or redcoats. Below the knee were knee high socks or boots. At the same time the sans culottes disdained this aristocratic style and began to wear ankle length trousers instead.
Culotte is simply the French word for "breeches", which nearly all men except the poorest wore, in many colors and materials depending on economic status, as you can see in the 1776 Les Costumes François. They were not inherently fancy or white. The sans-culottes were trying to represent their alliance with the poorest of their society by wearing pantaloons, although said pantaloons were typically better made than what the agricultural laborers, sailors, and beggars were wearing; period art shows them being fitted much like breeches.
I am curious as to what did dark age germans (who wore wool trousers) wear in deep winter, and also what did industrial europeans (who, thousands of years later, also wore wool trousers) wear in deep winter. For example, did russians in 1812 put on multiple trousers? If so, did each pair have its own belt? Did they make quilted trousers?
I would suggest asking these as a separate question, or rather as two separate questions - one on medieval winter clothing and one on Russians in the Napoleonic Wars - because they are getting lost in this post, and the people who can answer them are likely going to miss them due to the title.