We can experimentally verify spirits by communicating with them. Many such cases.
Now, do you mean measure them using our physical devices that we currently have? No, we can't do that. They are "minds beyond ours" as OP suggests, just not in the way that OP assumes.
Djinn: Demons. Santa Claus: Saint (i.e. soul of a righteous human). Demons: Demons. Fairies (real, not fairy-tale): Demons. Most spirits that you're going to run across as presenting themselves involuntarily to people are demons because demons are the ones who cause mischief. Angels don't draw attention to themselves.
I don't know, it seems reasonable to conclude that the experiments you describe point strongly to an endogenous rather than exogenous source of these experiences, especially since people who have these kinds of experiences do not all agree on what they are or mean and the experiences are significantly influenced by cultural norms.
An electron is a bit like a demon in the sense that you can't see one directly and we only have indirect evidence that they exist. But anyone from any culture can do an oil drop experiment and get values which aren't culture bound, at least in the long run. People have been having mystical experiences forever and the world religions still have no agreement about what they mean.