1. I think you just answered that question by saying it is attached to the rocking horse, so that would be the answer to go with.
2. In real life? There's no sure fire way to say, "This is a ghost". There's multiple tools that can be used though to try and establish some line of communication with them, for example: digital voice recorders, rem pods, dowsing rods, video cameras, EMF detectors/K2 meters, etc. I imagine, from a scientific point of view, it would be better be described as an unexplained phenomenon or some form of residual energy, though the latter is more speculation than something that is able to be proven.
3. Ectoplasm is fake. Mediums would use cheesecloth, gauze, and other substances during séances to fool the people attending them. It has be considered a hoax for years and there is nothing out there to support otherwise. ETA: That's not to say it can't be written about, but irl, it is regarded as being faked.
4. If you're trying to think of a way that the Foundation could have acquired this object, think about how stories of urban legends spread. Word of mouth. Eventually, that information could get back to someone who actually knows about the Foundation. Where I live, news stories are done every Monday that talk about urban legends or paranormal events, this is done by my city's largest news station, so it's safe to say a pretty big audience watches that. Consider that another possible route that could ultimately be explored. That's not to say that, in-universe, state governments wouldn't know about the Foundation, but that begs the question of how this one particular item got the state's attention to begin with. It kind of comes back to the same problem, in that sense.