As per natural selection, is it safe to assume that some species will have it genetically encoded so that they produce a certain 'perfect' rate of mutation so that they can adapt to an environment that changes quickly?
Another way to put this; does every type of organisms have roughly the same amount of mutations each time they reproduce (relative to the size of their DNA)? If so, is/can/could this be governed by natural selection?
Drake's rule
Let $L$ be the size of the genome. Let $\mu$ be the average per nucleotide mutation rate, then the genome-wide mutation rate $U$ is $U=L \mu$. There is a general relationship between the genome size of an organism and the nucleotide mutation rate that makes that the genome-wide mutation rate is often approximatively 1. There is of course several orders of magnitude of variation around this value of 1 among organisms.
This rule is called Drake's rule after the name of the author who first discovered it (Drake, 1991).
To my knowledge, this relationship is not fully well understood. Because you are looking for an adapted mutation rate in your question, I would like to highlight that I would not go about assuming this is adaptive. I could imagine that some molecular genetic constraint of the replication process could be causing this type of relationship between $L$ and $\mu$.
You will probably want to read this post about Drake's rule.
Mutation rate in humans
In humans, the average mutation rate is estimated to be around $\mu=2.5 \cdot 10^{-8}$ (Nachman and Crowell, 2000; see also this post), with a genome size of about $L=3 \cdot 10^9$, leading to an estimate of the genome-wide mutation rate of 80.
Interestingly, the mutation rate is in fathers is age-dependent. See the post Gender and age-specific mutation rate in plants.
