Would there be an overlap of the priesthood and researchers? How would
an average person of faith view scientific study?
Historically speaking there was a substantial overlap of ordained clergy and scientists. This is in part because of clerical training being the "obvious" route for intelligent people to get an education of any kind and no doubt also for many other reasons. These days I think both tend to be full-time jobs and therefore logistically difficult to combine.
Without any further information about your hypothetical world, there seems no reason to assume that the same trends couldn't happen if you want them to. And so yes, of course priests can do science if they choose.
As for how those of faith view scientific study -- in my opinion the general trend is that if someone believes something on faith, whether this is religious faith or some other belief not inspired by the scientific study of evidence, then they react badly to science that contradicts it. Occasionally, some people of faith might say that inquiry is of itself disreputable, but I think this is extremely rare. Many (all?) real-world religions hold "finding out stuff that we don't already know" to be a good thing. It's "contradicting stuff that we do already know" that causes tension.
So, if a hypothetical priest believes that the earth is forged from pieces of gods, and it actually is, and evidence of this is scientifically analysable, then clearly there's going to be no conflict there. But if the same priest is dogmatic from scripture that Europe is made from some god's left arm, and scientific inquiry finds that Spain is a gigantic right hand, then there's going to be a conflict between science and that specific claim in scripture. Suddenly this priest faces a crisis of sorts.
Galileo got into trouble with the church, for example, not because the church was absolutely opposed to the scientific method in all situations (it wasn't) but because those people were unable to reconcile heliocentrism with their faith [the story of how that trouble escalated is rather more complex]. This isn't an inherent contradiction between Christianity and science, since there are in fact many Christians (Galileo among them) who see no contradiction at all between heliocentrism and their faith. But the church at the time had stated dogmatically that Ptolemy's system was basically correct, and was not inclined to change its view quickly. In the view of other Christians it shouldn't have done that, and had no basis in Christianity for doing so, but that's what their religion meant to those people at that time. Religions that never offered any firm opinion on the subject have no difficulty with Copernicus's conclusions and therefore wouldn't need to persecute someone like Galileo who built on them.
Assuming a reasonably credulous and uneducated population, the "average person of faith" will therefore view scientific study the way they're told to by their priests, who likely will support investigation that doesn't contradict their faith and have difficulty with results that do. In your example, the faith being basically correct about the relationship between the earth and astronomical objects, is at a massive advantage. The Church of Galileo's time was demonstrably wrong about a number of points on which it had no scientific justification for even offering an opinion. Therefore inconvenient scientific results might be suppressed if they arise, but maybe they won't arise.
On the other hand, if the population is generally somewhat educated and has the means to draw its own conclusions, then those in religious authority will have a much harder time simply "pulling rank" over scientists. Especially if the state is secular and so on. Which is not to say it never happens, but when the reasoning of both parties is available to the general public it becomes harder simply to say, "trust me, I'm a priest, I know all about astronomy, and that heretic with the telescope knows nothing" ;-) Inconvenient scientific results might not be easily accepted, but they can't just be banned.