I think "He keeps lying, causing her further problems" sounds more natural.

- Further causing: the act of lying had follow-on affects that caused problems for “her” - Causing further: there were already problems and the act of lying increased the number or severity of those problems Both would be understood by native speakers (and most wouldn’t differentiate) as the meaning is clear his lying is creating problems for her

Both are understandable as furthering the defamation. The first is a little more natural.

Yes

Yeah, from what I see it's correct, and sounds pretty natural.