Answer: Usually, when paving a patio with stepstones or pavers, the process includes digging down and placing a layer of gravel below the stones, tamping it firm, then a layer of sand, tamping that firm, and then the stones are leveled and set into place very close together. Finally, sand is swept into the small cracks. You would also use a firm edging to keep them all in place. This creates a well drained, water permeable surface that should hold the stones steady without heaving during freeze thaw cycles. (In some cases, the stones are laid into concrete which has been poured like a slab, again to prevent them from moving.)
If you simply put your stones on the ground and added sand between them, they may heave and require resetting each spring. If they are rocking, you need to relevel the ground beneath each one and adjust it to accommodate the underside of each stone. You may be able to loosen the soil and add some sand under it and wiggle it to sit flat.
Plain sand will not harden on its own, no. But if you want to plant between the stones you would not want to harden it. Also, if you create an impermeable surface you will need to grade the patio to allow for drainage so water can run off the area in a direction that is desirable -- not toward your house foundation, for example.
You could try planting a sun loving groundcover that likes heat and well drained soil such as a small sedum or creeping thyme into the sand between the stones. If you do not plant something, you will have weeds beginning to grow there. Or, you could try planting grass and then mow over the patio area as though it were part of the regular lawn. If you want to grow grass there, you should mix some good quality compost into the sand between the stones because grass does not thrive planted in sand -- it dries out too quickly.
I hope this helps.
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