That would be very strange IMO. Biologists talk all the time about the relationship between mutation and adaptation, and there are complex efforts underway, right now, to continue to understand how these are related. We’re talking about a mutation, that results from a change induced during replication, when that change was subject to correction but somehow escaped. Each of these has a nonzero probability of contributing to adaptation, but so do meteor strikes, and we don’t call those “potentially useful conflagrations.”
That’s what I mean by “precise replication.” The proofreading and correction are built in. In fact, the DNA polymerase machine includes those capabilities. It’s really not about assuming anything.
Well, the best way to understand them would be textbook-level descriptions or Wikipedia. The key player is DNA polymerase, which comprises families of related machines with different roles and properties. In bacteria, there are at least 4 different kinds of Pol, the “main” one being Pol III. All are composed of multiple parts that assemble into the final machine.
These are proteins. Proteins vary. They vary because of mutation. In principle, every single amino acid in a protein can be changed to any other. In practice, some mutations will destroy the protein utterly, some will not affect its function at all, some will affect its function only under specific conditions/circumstances, some will reduce its function in a way that the cell can live with, some will enhance its function, etc. There is nothing unique about DNA polymerases when it comes to their identity as proteins that are encoded by genes that can vary.
FWIW, some DNA polymerases are “designed” to make mistakes. A family of Pols in bacteria, called Pol IV, is called into action under highly stressed conditions and its role is to indiscriminately induce mutations, i.e., it is error-prone. (It has even been called a “mutase.”)
You are wise to reflect on how your thinking is not always well suited to initial consideration of biological questions. The existence of mutation means this, among other things: when a bacterium divides and yields two copies of itself, there is a possibility that one of those daughters is different from the original, or that both of the daughters are different. It’s one thing to know this fact; I think it takes another step to incorporate these kinds of stochastic phenomena into your basic thinking about biology, evolution, populations, etc. And yet, without converting away from law-based physics and its reliance on “types,” you’ll always struggle to think about biology.
These are open and active questions, and there is no single answer. Standing genetic variation is a fact of any population, but there are interesting and complicated relationships between mutation rate, population size, standing variation, and adaptive capacity. Your question doesn’t have a single answer.