"The man hears the woman's daughter through the wall."
Translation:Manden hører kvindens datter gennem væggen.
It should work, since it is here used as a preposition and then both can be used. But only "igennem" can be used as an adverb, e.g.: Hun læste bogen igennem.
No, both translate into "wall" in English, but are quite different. A "mur" is generally not a part of a building, it's open at the top, often around the garden of a house or some other area. A "væg" is part of a room/building. The walls of a house are topped by the roof, the walls of a room by a ceiling.