My background is nothing to do with biology, but it has struck me for years that natural selection is just a broad statement about probabilities that can be applied almost everywhere.

It is something like: if something has high odds to exist, given enough time to possibly exist, it will exist.

So an inert protein coming into existence? Sure. Better odds to be replicated? Then it'll happen. We ascribe agency or will to survive. Those probably help the odds in a lot of cases. But the concept is broader than that.

natural selection applies to any self-replicating things... viruses are considered non-living, but they are still self-replicating.

(technically, the self-replication need also admit mutations into the replication process for natural selection to apply. viruses do mutate so, they qualify)

Viruses evolve in a way that's very similar to living things (and some scientists consider them living, but that gets into philosophical definitions of life). They have genotypes that can mutate, and some of these mutations find ecological niches where they reproduce better than others, which leads to those genomes reproducing themselves more, i.e. evolution by natural selection. They're taxonomized similarly to living organisms as well, into phyla, orders, families, genera, species, etc.

Natural selection is a consequence of mortality and reproduction, not life in total.

You could read Lee Smolin’s book about natural selection in black holes!