Does moving a Work Item to the bottom of one view, like one team's current Iteration Status screen, affect its single Rank value which is also interpreted relatively in whatever other pages or views it might appear?
And therefore that other users might suddenly see it as lower priority than before, in their own other views too, without anyone having intentionally changed it there?
Yes, it can cause a user to see the ranking as lower priority than before. Ranking is always performed relative to the work shown on the user's view. When you drag a work item to a new position, it sends Rally a command to rank that item above or below the work items in the new position. And if other users don't have access to that work item in the new location, it can appear that it didn't move at all, or if they have access to and are showing more work items, it can appear that the work item moved drastically.
Let's take the following scenario:
You have 2 projects, Project Alpha and Project Beta
You have two users, Adam and Brian. We'll say
So currently the rankings will be as follows
Adam sees this ranking
Brian sees this
Now, Adam moves DE3 above DE1 so Adam sees
Brian, doesn't have access to Project Alpha so what he sees is slightly different
Now, let's say that Brian wants to re-rank DE2 to be the highest-ranked artifact he can see, so he moved DE2 above DE3. Now he sees this:
Since we're ranking above DE3 Adam sees this:
So the ranking is set according to what the user sees on their screen at that time; whether that's by permissions or view. You could accomplish the same scenario above by setting up two Work Views for your account where you showed 3 defects on one and 2 on the other and ranked them while switching between the views.