Well, I think Nuka put it best when he said "Scar wasn't even his father, we just took him in." That should answer a lot. Scar wasn't Kovu's father, but I do believe that Zira told the truth when she said "He was hand chosen by Scar to follow in his paw prints and become king!" This says a lot about Kovu's past without shedding any light on it whatsoever.
As far as Zira's strength, she hit Kovu with her claw. It still does take quite a lot of power to cause a scar, but it doesn't take that much effort to break skin. And when she's dangling off the cliff, ask yourself this: Would
you be able to pull yourself up a 90 degree incline with no platforms to stand on?
But back to Kovu, I think it's good that they don't go into the history of his father. Leaves the heart to wonder what the mind hides. After all, the mind doth hide what the heart doth seek. And this quote of mine is also very adaptive to Kovu.

"A film should engage, not distract. expand, not contract. Inform, not retract. Redefine, not reenact."
Credit to my awesome signature goes to Twin.