1. Do you take the two oughts in "what oughts ought to be" to be in one case, an empirical ought and in the other, an ontological ought? Which is which?
Thinking as a philosopher: The first "oughts" would be empirical, and the second "ought" would be ontological.
Though, I am not sure if it is fair to pick apart the phrase "what oughts ought to be", because it sounds like it could lead to a bunch of semantic traps. It is more important to emphasize that all oughts reduce to a type of is, at some point. Even the ontological ones.
2. In the present context, what does it mean for a fact to be right or wrong?
It is objectively true, preferably from a scientific perspective (experimental results bare it out).
Are you using fact and premise interchangeably here?
I suppose I could be. I did not think about that very much. Perhaps a premise is a special example of fact, in some way?
If so, is a true premise the same as a right fact?
A 'true premise' would be an 'objectively correct fact'. So, the short answer, I guess, is "Yes".
3. Do you consider the conclusion in my counter-example to be sound (ie. premises are true and logic is valid) as far as it goes?
Actually, one of the premises does look faulty. Sorry I did not notice this, before. But, I took "proficient at killing" to mean something like "enjoys killing" instead of "good at killing".
I suppose the logic would be more sound like this:
Premise 1: Bob is
takes enjoyment from ending lives*.
Premise 2: Jim is alive.
Premise 3: Bob has an inclination to end Jim's life**.
Premise 4: In philosophical terms, this inclination is thought of as an "ought".
Conclusion: Bob ought to kill Jim.
(* The sources for that enjoyment would be facts in the matter: Perhaps a mental disorder, or he was brainwashed. In this case, proficiency would be largely irrelevant.)
(** Premise #3 now looks a little redundant in there, but I kept it in for completeness of the decision making process.)
Of course, a typical layperson would find this unsound, but more for reasons of disgust at its conclusion, rather than the logic or accuracy of the premises. The sources of that disgust would be a separate issue.
The science of morality would probably find out why Bob's view of reality is so distorted that he would enjoy killing people, and feels he ought to do so. And, from the perspective of well-being, his Conclusion would be a wrong one to have.
Of course: Science and well-being are things we have to value. But science can ALSO tell us how and why we generally value these things (perhaps from an evolutionary perspective).
So, in one way: those values also become facts of the matter, in the end.
And, in another: The existence of the "covert ought" is not relevant to determining the facts of Bob's brain.
4. How can we determine what is wrong with Bob? Can it be formalized? That is, what are the premises and logic to be considered here that were not included in the original examples?
I hope I covered this in the previous answer. In short, it can be formalized just as effectively as neuroscience and psychology can be formalized. Though, of course, we are now talking about my altered Premise #1.
In your original counter-example, it was implied that one should kill people simply because one is good at doing so. But the conclusion really did not follow from it: It lacked "motivation".