Well, it could just be that Davos (and co.) will go to Mormonts as we have seen in the trailer and Sansa/Jon will go to Manderlys if they decide to split. If not then it would make sense to go to Mormonts first since it's much closer to the wall and much faster to travel with a ship. But then again this is the show with Littlefinger teleporting all over the place so who knows.
True. Technically they shouldn't really be able to get to White Harbor but Littlefinger can teach them instant transmission.
Brienne will use it to get to Riverrun.