The marshmallows used do need to be the little freeze-dried marshmallows meant for hot chocolate. Regular mini marshmallows will melt during baking.
The leftover cookies can be stored in a food storage container or zip-lock bag at room temperature for up to five days or frozen for up to three months. If freezing the dough, I suggest not mixing in the mini marshmallows until you’re ready to thaw the dough to bake.