In "A Rose for Emily," what conflicts existed between Emily and her father? (For whom or what was the horsewhip intended?)
All that we know for sure in this story is that Emily's father discourages all of the suitors Miss Emily had, which is likely to have been a conflict. We know that he ostensibly did so because none of them was good enough for her, not being of the socioeconomic class of the Griersons. We are told that the townspeople
...believed that the Griersons held themselves a little too high for what they really were. None of the young men were quite good enough for Miss Emily and such. (Part II).
And we are given an image, what the townspeople saw in their minds,
...a tableau, Miss Emily a slender figure in white in the background, her father a spraddled silhouette in the foreground, his back to her and clutching a horsewhip, the two of them framed by the back-flung front door.
This is a striking image, of the father as a violent man, fending off all the young men come to court Miss Emily, with Miss Emily, small, thin, and helpless, purely dressed in white, standing behind her father, turning away from the father's brutality.
Why did her father turn away these young men? The story does suggest that they were not good enough for her, but another reasonable inference is that he wanted to keep Miss Emily at home to take care of him. Whatever the reason, it is likely that Miss Emily longed for love, marriage, and escape from her father. For a female in that time and place, marriage would have been the only escape route she would have had. But for any young women, in any time and place, such a father would have created a considerable conflict.
In "A Rose For Emily," the exact details of the conflict between Emily and her father are not revealed by the author. We can, however, make some inferences from the text. First of all, there was a conflict between Emily and her father on the issue of her romantic relationship and the possibility of her marriage. According to the text, Emily's father believed that "none of the men were quite good enough," and this quote is accompanied by an image of Emily's father standing in the background and "clutching a horsewhip." The reference to the horsewhip implies that Emily's father literally fought off her suitors, and this is confirmed later in the story when the narrator mentions the young men who had been "driven away."
It is, of course, possible that he used the horsewhip on Emily as a means of exerting his will. Whatever the case, the horsewhip symbolizes the balance of power between Emily and her father. He refused to relinquish his control over Emily, though his reasons for acting in such a way are never explored.
I think you're trying to read too much into the story. The narrator simply describes Emily's father as sitting "spraddled" with his back to Emily and "clutching a horsewhip." We're never given any indication that he was physically abusive to her. Yes, we can speculate about their relationship and imagine all sorts of scenarios. We can also speculate just as well that the whip was to keep outsiders away. Remember that the narrator says, "None of the young men were quite good enough for Miss Emily." When she is still unmarried at 30 years of age when her father dies, the narrator says that surely she wouldn't have turned down a proposal if she had ever received one. Add to that the fact that she has her first gentleman friend after her father's death, and we can easily say that the whip was certainly a tool to keep men away from his precious daughter.