Why does halite have perfect cleavage at the (110) plane? Why doesn't it cleave nicely along the (100) or (010) or any other planes? What determines its cleavage plane? I know it has something to do with the fact that it crystallizes in face centered cubic  system but I don't understand why things happen the way they do.
I think the commenters have identified the issue. Halite does not have perfect cleavage along the {110} plane. As recorded in Mindat, Halite has perfect cleavage alone {100}, {010}, and {001}. This makes sense, as Halite is a cubic mineral (fcc), these are the planes you would expect it to cleave on.