Yes, that's right. Much like "his" or "her" in English, -Daj is a very general reference and might refer to any of a large number of people. In both the English, "The warrior killed her enemy," and the Klingon, jaghDaj HoH SuvwI' it is not specified which "her" we are referring to. It might be the warrior or it might be some other "her". Without context, we don't have another "her" or "him" for it to be referring to, so I would assume that we are talking about the warrior's enemy. But in your example, you are giving a little more context and I would be more likely to assume we are talking about Kahless' enemy in a case like that. If you really want to make it clear, you can be specific about who's enemy they are: qeylIs jagh HoH SuvwI' "The warrior killed Kahless' enemy."
If by that you mean the placement of the subjects and objects is the reverse from where they are in English, then yes.
Since Duolingo has hidden the Tips & Notes I want to make sure you know about them and where to find them. If you have not been reading the Tips & Notes, I would like to ask that you review those so we don’t have to continuously repeat the information that we have explained there.
If you are doing the course on iOS or Android, you cannot currently access the Tips & Notes through the app. To access the Tips & Notes, you will have to access the course using a web browser at https://www.duolingo.com/. You can still do it on your mobile device, but you will have to use the web browser instead of the app (or you can do it from a computer). When you click on a Skill, it will expand to reveal a Start button, a key, and a light bulb.
If you click on the light bulb it will reveal the Tips & Notes and give you a detailed explanation of the grammar that is introduced in that Skill. If you have questions after reading the Tips & Notes for any Skills, then please return to the forum to ask your question, explaining what you didn’t understand or what seems contradictory to you.