I don't know much about Stoic philosophy but if you look at that question: "How can we have responsibility in the Stoic's deterministic universe?"
It implies that those who are asked have a free will. A deterministic universe and a free will are incompatible. If that question is asked to us, human beings, most of us would hold that individuals are accountable for their acts. Why? Because we are social animals that evolved in a jungle. We used to see individuals commit things and we used to hold those individuals accountable. We didn't bother going further than that. And, as long as, our species have been successful in surviving (so far), we must say that that approach was evolutionary stable (in other words correct for the sake of survival).
Now, if you ask me to imagine a deterministic universe and a free will in it, I'm not sure that's even possible to imagine. Because, a brain must be capable of predicting the environment it is in, to be called a meaningful brain. That means the environment influences the brain. And, a logical next step is to say that brain activity is determined by the environment it is in. If the brain doesn't predict the environment it's a detached brain. A creature with that kind of brain wouldn't last long, and even if it did, we wouldn't call it an intellectual creature.
But, of you ask me to imagine a brain that is good at predicting the environment it is in, and also has a free will, how do you even implement that kind of brain? Do you put a random dice in it? If brains decisions are chosen randomly, will it really be good at predicting (describing) the environment? Would we call such a brain meaningful?
As of nondeterministic universe... Every state in the universe is determined by its previous state, which in turn is determined by its previous state and so on. To imagine a nondeterministic world we, here too, would have to incorporate some random dice (assuming the dice is effectively random). Every state of the universe is now determined not by its previous state, but by a random dice. It's not difficult to see the level of chaos in that type of universe, wich would be incompatible with (potential) intelligent life in it.