Let's take human DNA as an example. It contains 3.2B GTCA base pairs. This gives rise to 4^3.2B possible combos. It's just not possible to navigate this space blindly. There is not enough atoms in the universe to do that. It is known that there is bias in what mutations are favoured.

Only a tiny percentage (around 1%) of the DNA in chromosomes codes for proteins.

And yes, certain mutations are favored precisely because of the chemistry constraints (an extremely basic one is which base pair changes actually alter the resulting protein; a more sophisticated one is which amino acid changes alter the physical functionality of the protein).

There are multiple constraints that I can immediately identify. Maximal temperature extremes, barometric pressure, atmospheric/substrate compositions, etc. The bias is inherent to the history of the planet Earth and the gradients present across that time and space. I'd say it's highly constrained.

Of course there is bias, the bias is provided by the natural environment where the organisms coded by the genome must thrive or die. The bias is applied after the mutation occurs, but the mutations themselves are random, or nearly so. Probably there is some differential rate between the likelihood of each of the four base pairs to mutate into each of the others, but I would guess its nearly parity, because that would probably be close to optimal (though that depends on the details of the genetic coding scheme, ie the triplet code that translates nucleotide triples into amino acid codons).

I don't think this is true at all. There are multiple sources talking about how the mutation rate is context-dependent.

But is the diversity really that staggering? I mean most animals including possibly dinosaurs that have ever existed share a lot of internal organs, in the same place. They have eyes, brain (with a lot of the same brain areas, even birds have something like a prefrontal cortex but it's called something different). They all have legs, torso, head. I would say there is a lot more commonality than difference. The differences come from slight variations on a basic template that works, and then the body looks different and so on.

I'm not sure how to think about the diversity that evolution creates and how diverse it actually is. I would say there are _a lot_ of repeating patterns all across history, with variations on those repeating patterns always changing.

You're choice of samples is rather skewed towards ones sharing a relatively recent common ancestor. Octopus and Sea Squirts are also animals, and they don't have legs or torsos or, in the later case, heads or eyes. Octopus brains are also rather different from those of vertebrates, and they have 8 mini-brains for more distributed/localized control of each major limb.

That said, I agree with you that there is a lot of commonality in life. Even in the case of Octopus we share a lot of DNA. I just mostly think that is due to common ancestor and common environmental pressures, not to some fundamental limit in the breadth of evolutionary potential itself. Its probably worthwhile to wonder at how that actually works though. Maybe evolutionary potential could be improved.