Do we have to commit sins because of another's karma

Read this https://en.wikipedia.org/wiki/The_unanswerable_questions

"The [precise working out of the] results of kamma (Karma in Sanskrit)"

There are explanations of different kinds of action, different kinds of result, different kinds of kamma -- for example here:

• Kamma A Study Guide by Thanissaro Bhikkhu

And what is kamma that is neither dark nor bright with neither dark nor bright result, leading to the ending of kamma?

It seems to me that C's killing A is "injurious" and therefore "dark" and therefore not "leading to the end of kamma" -- and so it is instead, as you asked, "making new kamma".

I'm not sure it's fair to say that it's "because of A's Karma".

I mean, perhaps A's kamma was part of it but it seems to me simplistic to say, "Oh it was all A's fault, C had nothing to do with it."

I guess that it was because of the kamma of A and C that they met at all; but then C had a choice (or intention) of their own, and also a predisposition based on their own kamma and training, i.e. to kill or not to kill.

And so it's not true that "we have to commit", which is what you're asking in the title.

I suspect there are societies or cultures, where the doctrine of "obligation" is assumed to be true -- for example, "I have to kill you, because your dad killed my dad", and so on -- but I think that's merely a social convention and belief, it's not some fundamental rule of the universe.

If I understand it right there's a section of the Bhagavad Gita where Arjuna is told (by the God) that he has an obligation, a duty, to kill others, because of his kamma e.g. because of his birth as a prince. In my opinion this is more or less completely contrary to Buddhist doctrine.

When I search for "predestination" for example I find doctrine like this:

Fundamentals of Buddhism: Kamma

... It is karma that explains the circumstances that living beings find themselves in.

Having said this much about the function of karma, let us look more closely at what karma is. Let us define karma. Maybe we can define karma best by first deciding what karma is not. It is quite often the case that we find people misunderstanding the idea of karma. This is particularly true in our daily casual use of the term. We find people saying that one cannot change one’s situation because of one’s karma. In this sense, karma becomes a sort of escape. It becomes similar to predestination or fatalism. This is emphatically not the correct understanding of karma. It is possible that this misunderstanding of karma has come about because of the popular idea that we have about luck and fate. It may be for this reason that our idea of karma has become overlaid in popular thought with the notion of predestination. Karma is not fate or predestination.

If karma is not fate or predestination, then what is it? Let us look at the term itself. Karma means action, means "to do". Immediately we have an indication that the real meaning of karma is not fate because karma is action. It is dynamic. But it is more than simply action because it is not mechanical action. It is not unconscious or involuntary action. It is intentional, conscious, deliberate, wilful action. How is it that this intentional, wilful action conditions or determines our situation?

According to my understanding, when C kills A, C acts with intention rooted in delusion (moha) and hatred (dosa) and that act itself is/constitutes karma (intentional action), the results (phala) of that action will be bad if they ever ripen and produce their result.

As @Remyla pointed out, the workings of karma are said to be very complex and difficult to fathom and, as @ChrisW pointed out, they are typically not thought to be simple cases of a 1:1 returning of an action in the same way in a future life. So, the precise relationship between A killing B and C killing A is not clear but doesn't change the above.

Finally, my understanding is that one cannot be compelled definitively to take any actions by the actions of another. If that were the case, it would be impossible to escape from dukha by intentional action.

I am assuming we are not talking about accidental killing, but rather murder.