Yes, those are exactly the types of things I want to do, but can't. I click on the snapshot, and as soon as I move the mouse I get the crossed-out-circle "prohibited" sign.
I thought there must be "rules," like snapshots can't be moved if they're created from certain types of items/pages/stuff, or if they're created under certain circumstances, because sometimes things move and sometimes they don't, and it seems entirely random to me.

Sorry I was not clearer. You will get the prohibited sign when you are moused over places where you cannot drop it. So just to make sure things are working select a snapshot and drag it on top of a book. This should make the snapshot a child item of the book. Does that work?

To clarify -- I want to make the snapshot a child item of the newspaper article, but that's what it won't do. Won't move to a child item of a generic item, won't move to a different collection, simply won't move anywhere.

I'm sorry I was not more clear. You are right that there are rules for where you can drag a snapshot. Before I go into the explanation I will offer my best guess at what your issue is. I would hazard to guess that you are trying to drag a snapshot onto another snapshot. That the newspaper item you are draging the snapshot onto does note have the little newspaper icon next to it but the little camera icon. If this is the case you can manually create a newspaper icon and drag both snapshots onto it. If this is not the case we probably have a bug and I would ask you to try out the other behaviors I list for snapshots below to see if you are having other issues.

For the rest of this I will refer to the snapshots as attachments to offer more generalized explanation. Links, snapshots, PDF's, images, or any type of file you store inside Zotero follow these same. Thus there are really only two kinds of items, reference items and other files (attachments).

Similarities:
Both kinds of items can be top level items. You can drag any attachment out into the collection it is in. it will then appear on par with your reference items.

Differences:
Reference items can have attachments, files cannot. Thus you cannot drag snapshot onto a PDF, or a note onto a image. The only kinds of items that can have attachments are reference items.

There is one final key note. When a file is attached to a given reference you cannot drag it directly from that reference into a different collection as a independent item. You need to first separate it from the reference by dragging it out into the collection it is already in and then you can move it to a different collection.

Sorry for the confusion, please ask me to clarify if anything is unclear.