Can sauerkraut be prepared with open lid? If the cabbage is successfully kept with weight, under the brine, all the time. 

I use a glass paper weight as weight. Can mold also form on the weight, on the portion which remain above the brine while using a closed lid or an open lid?

Thanks.
Making sauerkraut does not require a lid.  As long as the cabbage is submerged (and you have the proper brine) it will be fine.  Any mold that forms on the surface or the weight can be removed.