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I read Code by Charles Petzold earlier this year and I feel like I had a similar realization. A simple switch can lead to such complex systems.

Right. A transistor (as used in most digital logic, anyway) is just a switch, and is what our most complex computers are made of.

Do we have an understanding yet of the minimum number of unique rules a system must have in order to be able to be Turing complete?

Not sure how this translates to "rules" in a cellular automaton, but: https://en.wikipedia.org/wiki/One-instruction_set_computer

This is exactly what I was asking about. Thank you!

There's also a few "one instruction" CPUs

You can build anything with AND and NOT gates or OR and NOT gates.

You can build anything with just NAND gates.

Well, we know that rule 110, which is like a 2d versions of Conway's game of life, is also Turing complete. So at least that simple.

Growing Artificial Societies is an interesting book that extrapolates on this.