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Does this include damage tests with said stat? Example: 4th Circle Warrior attacks a horror, hits and rolls damage. Can he spend a point of Karma on the damage test since it involves his STR?

Nope, the ability to spend karma is discipline dependant and generally at circle 7

Not only are you wrong (as far as 3rd Edition goes, at least), but you didn't address the question at all.

Well I will concede the grammar in my response was a bit off, I was a bit rushed.

But my answer wasn’t wrong, incomplete in the point I was making but not wrong.

The answer is indeed `Nope’ when capable of spending Karma on strength only rolls, it does not allow you to add it to damage. As Arma has also stated.

For those few disciplines that do get to add Karma to damage it is generally at 7th circle, for example a warrior can spend Karma on close combat damage at 7th circle, and archers can add Karma to ranged damage at 7th circle.