"The girl puts the book down on the table."
Translation:Jenta legger boken ned på bordet.
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.
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