Cross country time is logged per whatever your applicable category is (61/141 or 91).

With regards to landings, the usual rule that I've seen followed is if the CFI is "guiding" the controls and assisting the student in inputs, the CFI logs the landing since they were manipulating the controls (they were not the only or main manipulator though, if they were, no question) or if the CFI assists on all landings, they sometimes split the total # of landings (usually in a 70/30 manner for the student). Likewise, if the CFI is sitting there because your student is fully competent enough to land on his/her own with no guidance, then the landing goes to the student.

61.51 is clear about everything except landings.