Xie Chongjin is on the side of the 2nd prince (kinda a given since his daughter is married to said prince). Not only does the second prince dislike the MC and her family, but it's also obviously to pretty much everyone that even though the MC's family never acted/mentioned support for the crown prince, their lifestyle/titles/positions are completely dependent on the emperor liking them. Since the crown prince is known to be fond of the MC and kind to her family while the second prince dislikes them, it's pretty obvious that things would be much better for the MC's family if the crown prince inherited the throne. So though they never said or did anything to partake in the power struggle between the first and second prince, it's pretty much an unspoken given that they would prefer the crown prince become the new emperor over his brother.
Yes. He does so in order to prevent the second prince (and the royal family) from getting dragged into a scandal and as a way to suppress the Shi family. More specifically, putting the blame on SFX and only her (that is, on a young daughter of the Shi family who isn't married rather than someone with any power, aka the patriarch), they change the narrative of the attack from one involving the battle of the throne into the act of a petty, temperamental, and spoiled young girl who acted out of jealousy. While still a severe crime, given the attempted assassination after all, it becomes the act of one person whose interests doesn't go beyond the boudoir, rather than what will pretty much amount to a conspiracy or power grab by one of the royal princes. Since girls aren't seen as important and seen as emotional dead weight most of the time, the "weight" of the crime lessens somewhat and the implications of it are much much smaller than if the truth was known. While the crown prince is married to the oldest Shi daughter, she (and he) are insulated from much of the fallout/implications of the case by the fact that...well, the case was reduced to the action of a young spoiled girl whose entire world was the boudoir and couldn't handle the boy she liked hanging out with another girl.