You can store shredded zucchini in the fridge for short term storage and freeze it for longer term storage. If you freeze them, you will have to put them in freezer bags so they last longer.
If the avocado isn't cut open yet, you can store it either in the fridge or out; you shouldn't store it around other fruits, because this will make it ripen...
If you've frozen some shredded zucchini, you could use it when you make vegetable soup. Try adding frozen shredded zucchini to a vegetable lasagna or cottage pie. Try zucchini, cheese...
Yes, you can store an excess of the paste in the refrigerator. Unless you freeze it, I would use the refrigerated left-over paste within three to four days.