I read somewhere that even a single small nuclear exchange between countries across the planet from us would cause a Nuclear Winter, which would kill much of the flora and fauna on Earth.
No, it wouldn't. Nuclear winter from a small-scale exchange was a simulation result many decades ago, to be disproven only a few years later by more advanced simulations and observation results from volcanoes, forest and oil fires. Larger exchanges will have a noticable effect, but how large is up for debate.
This is sensationalist nonsense, as evidenced by "just 100 warheads" without giving any yield or burst height. And if you read the Wikipedia article I've given you, the evidence points to the effects of any exchange in the 100MT range (which is far more than one would expect from e.g. India vs. China or UK vs. France) will be noticable, but far from catastrophic. At worst, it will cancel out global warming for a year or two.
Kyle Hill is a sensationalist? Richard Wolfson and Ferenc Dalnoki-Veress are sensationalists? That is an accusation I have never ever heard sent Kyle’s way. Did you actually watch the video or read the article in the MIT press? Why should I believe you and not the authors at MIT?
https://en.wikipedia.org/wiki/Nuclear_winter has a fairly detailed elaboration on the various points of view and their history.