Are genetic algorithms considered to be generative models?

Within the broader field of ML, I think generative models has a more specific meaning, and if someone says the phrase, I don't really think of GAs as what they were referring to. So in that regard, no, I don't think GAs would be considered generative models (unless perhaps you built one to do very specific tasks that we tend to use other algorithms for today that we call generative models). However, it's probably one of those things that any attempt to rigorously define the term in a way that reflects that based on some underlying principle is really hard, and you'd probably end up with a definition that very well might include GAs. Basically, I think we're in "is a hot dog a sandwich?" territory here, which can be a fun discussion, but doesn't really provide much in the way of useful outcomes.

In terms of specific questions, GAs don't generally have separate operators for "hone in on a new region of the space" and "generate solutions within that region". The GA has some genetic operators that when iterated and combined with selection, serve to move the population around the search space over time. All you're doing is generating new individuals via those operators. Humans interpreting the results will say things like, "that mutation caused it to find a promising new region to explore" or whatever, but the algorithm just mutated an individual using the same algorithm it always does. It's us imposing descriptions on the outcomes that provide the kind of color you're asking about.