I believe a lot of the "wintery" imagery we associate with Santa Claus comes from Woden/Odin in Germanic pagan tradition. That was merged with St. Nicholas.
Yule is also pagan Germanic in origin.
It's difficult to pinpoint exactly what influences went into the emergence of Santa Claus. Of course there's the apparent Dutch influence of Sinterklaas, but it's not a concrete connection beyond the similar sounding name because the Sinterklaas personage doesn't seem to have been very predominant with the Dutch in America.
It appears that a number of unconcious influences may have gone into the idea of Santa to suit the citizenry of Upper Class New Yorkers of the early 19th century as they sought to reform how Christmas was generally celebrated. Santa didn't exactly emerge whole cloth out of nowhere, but was fleshed out throughout most of the 19th century. Indeed there were a number of interpretations (as there still are today), but many of the most widely accepted ideas around him were pretty much established by the 1860s.-1870s. Clement Clarke Moore pretty much cements the idea of Santa travelling by sleigh pulled by eight reindeer and arriving on Christmas Eve (although Santa had already been depicted a few years earlier travelling by sleigh but pulled by a lone reindeer). Thomas Nast some forty years later establishes the idea of Santa being based with his workshop at the North Pole and having a means to watch children from afar. His generally accepted appearance began even before Clement Clarke Moore, but is more concretely fleshed out by Thomas Nast's illustration. And that general appearance would be refined (or more accurately widely popularized) some seventy years later by Haddon Sundblom.
And throughout the 19th century he did retain some religious connections. Churches were known to use Santa as a draw to get children, youth and others into the Church around the holidays. He still retains something of a religious sensibility about him in the sense that even as he became the ideal promoter for retail business he still managed to remain rather anti-commercial and almost Christ like in some of his qualities. His (usually) red and white attire retains a connection to the red and white in the rainment of St. Nicholas just as easily as it can be seen as a connection to the red of mistletoe berries and pointsettas as well as the white of winter. Down through history people reinvent or reinterpret things to suit themselves and often don't document the reasons why they made the choices they did or what influences went into their decisions.
Part of the appeal (even beauty) of Santa is he has managed to retain some Christian like qualities even as he isn't associated with any particular denomination or sect. On the other hand because he isn't owned or trademarked by anyone he can (and has) also be used to promote all sorts of things from toys to hardware to luxury items and even alchohol, cigarettes and guns. Santa has come to embody (from rather early on) the very dichotomy of the holiday season.
While there are some who decry how the focus on Santa seems to detract from the focus on Christ it's quite apparent most people apparently have no problem with the two coexisting.
As for the Christmas Tree it, too, (like the holiday season) could have its meaning changed over time. It could conceivably come to be seen as a religious symbol or have religious overtones in decades to centuries to come. Note how many churches choose to have a Christmas Tree within. And take note of how many people display a nativity scene under their trees.
I don't think the symbol seen in the
Enterprise's chapel is supposed to specifically represent a Christmas Tree, but it's not impossible that an evergreen (or Christmas Tree) could come to be seen as a significant symbol over the centuries into the future.
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