Shalloy,
The bulging / slumping is not unusual in a blue -- generally they are made with a very soft curd. For that reason, I actually return to the mold or put the mold over the cheese during the salting phase (i.e., apply salt, then put mold over) -- this will keep it from slumping too much.
The salting will draw moisture out of the cheese, so you should expect it to be wet at that point. Eventually the salt is absorbed, and it begins to dry. Note that this drying effect helps the rind to form.
I've only made a couple of blues, so take all of this with a grain of salt ...
As best I remember, I left my blues on the counter through the salting phase, and then put them in the cave, in a ripening container. However, I took them out for half an hour or an hour each day to dry a bit -- which is to say, yes, it seems like they stay very wet. I wiped out the container and then returned to the cave after the half/hour.