This comment https://news.ycombinator.com/item?id=16027737 about how to get around NP-hardness made me go "Oh, can you even ask such a question? Can you expect to solve getting around this kind of problem?" and so I went researching.
I found the paper Complexity of Protein Folding https://www.gwern.net/docs/biology/1993-fraenkel.pdf and chapter 4 exposes two possible realities: either nature solves NP-hard problems in polynomial time or it functions in a classic Turing machine model. I was like "Welp I guess when you put it like this, talking about nature working within a machine model I suppose it sounds rather silly, yes. It may be that nature just does her thing in her own way somehow." The paper says a protein with 100 amino acids may assume 8^100 conformations and nature solves for that in 1 second. Also the spin glass https://en.wikipedia.org/wiki/Spin_glass model is apparently some other thing that nature solver eerly fast.
Can we chain these natural occurances in order to calculate a function with inputs in record time?
is not factually correct. In reality, proteins never reach their lowest energy state unless they are very simple proteins, instead, they rapidly reach a state of kinetic interconversion between several accessible states. We can simulate proteins and measure those state transitions.
Really, solving protein folding is about classical approximations and clever sampling.
There are some problems with this - firstly that paper is quite old. Not sure what the current state of the art is but this might be helpful : http://folding.stanford.edu/dig-deeper
Secondly, the idea that the protein might adopt any of 8^100 (or some large number) of combinations is not quite right. Consider a very simple model of just a chain of nodes connected together on a 2D grid. There are certainly a large number of possible ways to arrange this chain on the grid, but if you start off in one particular arrangement (call it 'unfolded') and then move towards a more compact one ('folded') then you will follow some particular path of arrangements.
However, there are ideas around 'quantum biology' from Johnjoe Mcfadden (http://www.johnjoemcfadden.com/) that might be interesting.
As for your question: NP-hardness refers to the worst-case performance, not the average case. Many individual instances of NP-hard problems are solvable. Stochastic algorithms do quite well, and protein folding in nature is a kind of stochastic algorithm (classically we would call it simulated annealing, although really it is quantum annealing). So as far as we know, there's nothing nature is doing to solve NP-hard problems faster than known algorithms. But it does do quantum annealing very fast, something we would really like to exploit using quantum computers.
For more like this - the intersection between computability and nature - I HIGHLY recommend Scott Aaronson's "Quantum Computing Since Democritus".
That's essentially what a quantum computer does.
Molecular mechanics, which is interested in simulation of big biomolecules and the like, ignore quantum mechanical effects, or use a hybrid of quantum/classical mechanics, because the structures of interest are large enough the QM effects can be approximated away.
Other subfields of computational chemistry use only QM, with some approximations to make the programs tractable.
I can't think of a particular example for an unknown reaction being predicted first, then verified, but it wouldn't surprise me. However, there are cases of computations being used to guide experiment (who have to have an idea of where to look for something).
Theory is generally used to explain, rather than predict. Ie, given a reaction, what is the most likely mechanism? Or helping explain the source of a particular property.
With increases in computing power, databases, etc, we may see computational chemistry start to predict novel materials (via machine learning, for example).
A Compositional Framework for Reaction Networks
https://golem.ph.utexas.edu/category/2017/07/a_compositional...
https://news.unm.edu/news/unm-chemists-provide-theoretical-i...
The author info from that link:
Steve Carr
Supervisor, Communications
University of New Mexico
Big molecules are still a challenge because the scaling of the accurate methods is very bad (N^4 or worse, where N is the number of electrons!). These calculations are something that quantum computers could theoretically do much better than classical ones.