My guess for why Isagi is so obsessed with cleanliness and order is that she either started off as a germaphobe and had her phobia compounded severely by having a close relative/friend suffering a lot from an illness and/or unsanitary conditions in her early childhood, or she acquired the germaphobia from such a relative/friend suffering so.
As for why she rejects Komi... it's not that she's actually rejecting Komi, it's that she holds Komi (or rather, the Komi that she and everyone else other than Tadano, Manbagi and Komi's family sees) as embodying the paragon of both order and purity that she aspires to be yet falls so short of, and believes herself to be too disgustingly filthy to dare associate so closely with Komi.