The Buddha does not admit to a self AFAIK, therefore the presumptions your question are based on don't have a great base to start on. However, conventionally , this can be said in regards to your question.
1) What does the Buddha say about agency for our state of mind and our behaviour? Where does agency for our positive/negative emotions lie?
Someone, having not followed and understood the Buddhas teaching, regards the hot ball of iron in their hand as themselves, not knowing to put it down, they create their own suffering. When someone offers them another Hot ball of iron, they gladly accept it, and when it burns their other hand, they blame that person for their suffering. It certainly isn't wholesome to offer to hand such a thing to someone else, but it is out of ignorance to the way things are that someone should choose to hold onto it.
"In the same way, an uninstructed run-of-the-mill person regards form as: 'This is mine, this is my self, this is what I am.' He regards feeling... perception... fabrications... consciousness as: 'This is mine, this is my self, this is what I am.' If he walks, he walks right around these five clinging-aggregates. If he stands, he stands right next to these five clinging-aggregates. If he sits, he sits right next to these five clinging-aggregates. If he lies down, he lies down right next to these five clinging-aggregates. Thus one should reflect on one's mind with every moment: 'For a long time has this mind been defiled by passion, aversion, & delusion.' From the defilement of the mind are beings defiled. From the purification of the mind are beings purified.
2) On whom (or where) does the onus/responsibility for changing a negative situation lie?
The solution for such suffering lies in following the teaching of the Buddha, following the Noble Eightfold Path, and developing wisdom and understanding into the nature of phenomenon and described in the Satipaṭṭhāna Sutta
“Bhikkhus, this is the direct path for the purification of beings , for the surmounting of sorrow and lamentation, for the disappearance of pain and grief, for the attainment of the true way, for the realisation of Nibbāna—namely, the four foundations of mindfulness.
3. “What are the four? Here, bhikkhus, a bhikkhu abides contemplating the body as a body, ardent, fully aware, and mindful, having put away covetousness and grief for the world. He abides contemplating feelings as feelings, ardent, fully aware, and mindful, having put away covetousness and grief for the world. He abides contemplating mind as mind, ardent, fully aware, and mindful, having put away covetousness and grief for the world. He abides contemplating mind-objects as mind-objects, ardent, fully aware, and mindful, having put away covetousness and grief for the world. "
3) What do modern-day gurus say about taking the onus or alternatively, putting the onus on external agents, whether human or otherwise?
Many people say many things. Unless a teaching contains the Noble Eightfold Path, it will not produce noble disciples:
“Very well, reverend Sir,” the wanderer Subhadda replied to the Gracious One, and the Gracious One said this: “Wherever, Subhadda, the Noble Eightfold Path is not found in a Teaching and Discipline there a true ascetic is not found, there a second true ascetic is not found, there a third true ascetic is not found, there a fourth true ascetic is not found.
4) Is it correct to say to another person, "Unless you exercise right-speech, I am unable to have right-speech, right-intention and right-mindfulness?
No. see the answers to question 1 and 2