To answer the title: Mods have 3 options. Mark the report as good, as bad or ignore it. AFAIK neither options requires them to actually take an action based on the report.
I see some of my reports are just sitting there for more than 48 hrs or something, are they marked good but no action is being taken?
They might, but I doubt that. Its more likely that the report is either ignored or marked as bad
Another question, what if I report a post where it is displayed that a mod has moved the thread, if I feel the moving of thread is incorrect?
You mean the "MOVED: <title>" threads some mods create? If you think it was moved incorrectle I would report the thread, not the "MOVED: ..." notification that is was moved.