Many of your questions are actually not all that strongly tied to physicalism. So it may make sense to dispense with that first:
If physicalism holds, how fundamental can consciousness be?
It can't be. Physicalism states that everything supervenes on the physical world, so it can't be fundamental.
Now the rest of your questions deal with things like object persistence. This is a problem faced by every philosophical group, not just physicalism. It also deals heavily with the gap between empiricism and ontological truths. It also has interesting linguistic aspects:
Would it make sense to talk about individual discrete consciousnesses, localized in space and time, and claim there is one consciousness in body X at time t, another in body Y at time t, and so on?
Nothing would be inconsistent here. Nothing precludes making such a statement unless one already believes in an existing system which states that a consciousness in body X cannot ever be in body Y, for some reason completely separate from the question at hand. Indeed, we do see people who talk this way. There are people who speak of "past lives," so presumably it makes sense to them. The real question is probably what one intends to do with the concept, and whether the argument one makes with this concept is rational or not.
If it is possible under physicalism to say that the same consciousness exists at different spacetime locations under the right physical conditions, would it then be plausible to consider the concept of "reincarnation"?
Sure. We can define "reincarnation" to be this. Just be careful because "reincarnation" is a word that has implications for many individuals who use it, and those implications may or may not match with your definition.
Conversely, if physicalism suggests that it doesn’t make sense to track the identity of consciousness over time, wouldn't this imply that "my" consciousness from 1 second ago was destroyed, and "my" current consciousness, aware of my hands and the screen in front of my eyes right now, is a completely new consciousness with no fundamental ontological identity to the consciousness from 1 second ago?
As for this question, I would recommend the SEP article on Temporal Parts. It covers useful terms like "perdurable" which can be used to make very explicitly clear statements. What you describe is an "endurable" view that views time as a series of independent snapshots with no connections between them. Contrast that with a perdurable view which sees objects that "persist" over time. This distinction is independent of the discussion of physicalism. You can make endurable arguments in physicalism; you can make perdurable arguments in physicalism; you can make endurable arguments in other systems (like dualism); you can make perdruable arguments in other systems (like dualisms).
To close out, I'd like to add two unsolicited pieces which I think may be of use. If I may read between the lines, I get the impression you are not a believer in physicalism. An exercise I have found very useful is to attack one's own beliefs with the same ferocity as we attack a belief that is alien to us. I find it a useful exercise that often provides useful fruit. Often I learn more about my own beliefs by taking a challenge I gave, applying it to myself, and realizing the same troublesome quirk applies to myself as well! I don't know what beliefs you hold, beyond the ones you assert here, but see what happens when you shake them!
The second unsolicited advice is to explore what it means for a body to be "the same." I do get the impression you have an intuitive sense of what that phrase means to you, and what you are seeing clearly doesn't sit well with the arguments for physical ism you are reading. Consider what constraints might be needed to be added (or removed) from the concept of bodies being "the same" to make it more palatable. Our bodies are not isolated. They're part of a greater system. Quite often we find that its hard to separate a body from its environment cleanly. Physicists have the concept of "boundary conditions" which helps somewhat, but with multiple interacting systems, there is often emergent behavior which can drive necessity in ontological models.
Since the Pope believes that a human soul is linked to the human ...
- How is the Pope relevant here?