I would probably let it set overnight, or for at least 8 hours between each coat. For a finer paper, use strips of pages from an old phone book. You have a week, so it sounds like you have plenty of time to do about three-four layers. Just mix up the mache in smaller batches. For the bottom, is it supposed to sit flat? Pop the balloon after the first batch is dried. Then cover the bottom with a piece of cardboard so it's a flat surface. Tape the cardboard to the sides of the boulder, and cover the bottom with the goopy strips. Can you set it on something like a football tee, so it gets air all around? Then you can do the whole thing at once. Have fun.

Something that works great for the outer layers after the structure is formed is to use pulp from cardboard egg crates.
Just rip them up and cook for a while with just enough water to cover. when the pieces soften use an old hand blender or food processor to break up the pieces. The pulp will be sculpt able and You won't need to add glue to it. This makes it easy to add details and texture plus its a nice grey stone color.