A lot of modern day Santa Claus mythology is somewhat borrowed from the Norse god Odin (a bearded wanderer who enters homes through chimneys; the eight reindeer are reminiscent of the eight-legged reindeer Sleipnir, etc.). That's all well and good, seeing as lots of Christmas borrows from the ancient Norse/Germanic holiday Yule which may have been worshiping Odin.
But also Santa is St. Nicholas. St. Nick was an actual guy who lived in southern Anatolia, was the patron saint of sailors and punched a guy in the face at the Council of Nicaea.
I get why they wanted to replace Odin with someone less pagan-y (they were Christians now, after all!), and with the Catholic tradition of venerating saints, it makes sense that they went with a saint. But why didn't the northwestern European tradition that gave us Santa pick a northwestern European saint? Why not just, like, turn Odin into a saint ("Saint Odin") in the same way that the Irish turned Brigid into Saint Brigid, or make up a possibly-fictional saint like I think what happened with Saint Valentine? Why pick Saint Nicholas instead of any of the many, many other saints?
I'm citing u/KiwiHellenist a lot recently (they're the poster who tends to answer these particular questions) but see this thread for discussion of the "Santa is Odin" thing.
Broadly the issue here is that the question is backwards. St Nicholas is the original Christmas-gift-giving figure, and was never based on Odin at all, so "why was St Nicholas chosen to replace Odin" isn't really a questin can be answered.
You have already received a reply about your main question, so I hope it is okay that I reply to two things from your rather long body of text.
Eight reindeer = Sleipner: The idea that Santa uses reindeers comes from an American 19th century poem and that it was eight of them comes from another American 19th century poem^(1). So this tradition is not old enough nor universal enough to have any connection to Odin. It is also worth noting that the Odin - Chimney connection is rather difficult to find any proof for.
St. Nick punched a guy: As u/talondearg discuss in this thread is there no ancient sources for this claim. It seems to all come down to a 14th century legend and has no connection to the real St. Nicholas.
1 https://archive.org/details/santaclausbiogra0000bowl/page/40/mode/2up
Thanks to /u/This_Rough_Magic for the summons.
There are a lot of good discussions here together with links that should be helpful.
Just a word about how there can easily be a flawed perception about the source of a folk tradition. Our modern, scientific perspective seeks to define "the" source of a folk tradition. The fluidity of folklore does not usually work that way, and traditions never behave themselves!
Many factors can become part of the tangle of threads that yield a folk tradition, and these, in turn, can be influenced by many other factors. Despite this, various writers have, from time to time, announced that they have "discovered" the source of some motif. Sadly, "discovered" usually means speculated, likely while taking a shower.
Santa Claus is what folklorists would refer to as a fict: something that non-believing adults describe to children with the intent to be believed (the word was coined by the great folklore theoretician, Carl Wilhelm von Sydow (1878-1952)). Compare the tooth fairy, the stork, and the Easter bunny.
Many threads went into the making of the modern image of our beloved jolly fellow, but since the stew was simmering in the kettle for centuries, almost entirely in shadows without illumination of the written record, we can only guess at how things before a few recent decades contributed to what we have today. This is key - because those who say they know the source of Santa Claus are only guessing.
The final chapters of this process, beginning in the early nineteenth century are more easily observed in primary sources - everything from the images of Thomas Nast to the ways Coca-Cola added a more recent influence. None of these, however, are "the" source of the modern Santa Claus. Each had an influence, but again, it is a tangle of threads. None of these factors, however, "created" the modern Santa Claus. They each contributed to the constantly evolving and shifting motif. This is not a scientific, chemistry experiment where heating x with y yields z.
Did Odin and his eight-legged horse (not really a reindeer) transform into Santa Claus? That seems a reach to me, ... but!!! ... it is not impossible that as various threads tangled through time, Odin somehow had his own effect. The treads as they tangled were buffeted by the winds of conversion and augmented by the peculiar holdovers that may have become misunderstood and/or misremembered, but this is a process we can only imagine. There is no documentation of how these factors passed through time.
The paths (very definitely plural!!!) to Santa Claus include the Christian saint (as celebrated in Northern Europe), the tomte and other household spirits together with the dead ancestors, all of whom emerged in a special, powerful way to visit the hearth at the winter solstice, and "other factors" - to cover all the things known and unknown to us. All these things contributed to the emergence of the jolly fellow, who was finally shaped in more recent times by commercial, literary, artistic, and other influences.
Now excuse me; I need to set out some cookies to make sure that Santa feels welcomed when he arrives.
edited because I needed to add some cinnamon and nutmeg to the cookies. Best wishes for the holidays to all!
Another question, how did English speakers end up calling him "Santa" (Spanish) Claus/Klaus (German)? I would expect English speakers to stick with "Saint Nicholas"....or maybe "Saint Claus".