Putte will be very unusual to say when it is about laying something on top of something, it is used about "into", when the object disappears from view in a container of sorts. You can put something in your pocket/bag/drawer. I think you can put books in a shelf too. "Put it down" would be Legg den fra deg.

Setter and sitter have more of a vertical meaning while legger and ligger have a more horizontal meaning So you can legger a book on the table, because i guess you lay it flat ; but you can

I didn't get this question with a hover like you did, so I can't exactly answer your question. But if it serves as any help, I can explain when to add the E and when not to. When you're stationary, or you don't want to imply movement, you use it [the inflection]. Jeg bor der nede (I live down there). But when you do want to imply movement, you take it off. Jeg kommer der ned (I'm going down there). Hopefully I didn't get this wrong