Since the ceiling in the attic is too low to make it into a living space, you should only insulate the floor. Insulating the ceiling will create a new heating zone, and essentially you will be paying more to heat your attic.
Another reason you wouldn't want to insulate the ceiling rafters is that you can wind up with ice dam issues. The roof could get too warm and cause the snow to freeze more rapidly than normal and then refreeze into a sheet of ice when the weather gets colder such as in the night. This melting/freezing cycle could eventually let ice seep under your shingles and enter your home. It is a very common issue in homes that are in cold climates.
You should top off the insulation in the floors if it is cellulose, or replace it with new fiberglass batts if what is there isn't adequate. If you want additional insulation, you can cover the entire floor of the attic with rolls or batts of unfaced fiberglass insulation. When you are done, you should have a "carpet" of insulation on the floor. When selecting insulation, also look for the highest 'R' factor you can find in your area.
As far as the storage goes, you can suspend some shelves from the rafters by chains to avoid disturbing the insulation. If you do this, avoid storing heavy objects from them such as air conditioners etc. The shelving should be adequate enough to store boxes of holiday decorations and other seasonal items. Just be aware that the items up there are not going to be in a conditioned space, so don't store items up there that are sensitive to heat or freezing. I would also recommend storing any of these items in plastic storage bins to help deter any critters from nesting.
I definitely would not encourage children to go into the attic. There are many dangers up there which you would not want to subject them to. Depending on your attic there could be exposed wiring, poisons, toxic chemicals and fumes, etc. They can also get splinters, or be cut by exposed nails.
There are products on the market which can seal off the whole house fan for the winter. Since the attic is not a conditioned space, there probably isn't much of a need for it.
You would also want to look around your home for air leaks which lead into the attic. Access panels, etc. should be sealed for the winter. You can also use expanding spray foam in a can to seal up any gaps you find.