I would pull up the layer of plastic under the rock. They put that stuff down thinking weeds won't grow through it, but the dust blows in and the weeds grow on top of it. Unfortunately grass will not grow on top, at least not very well, and not in most of AZ.

Need to know where you live, name the town, because it makes a difference if you live in the high desert, low desert, or the mountains.

Whether you need soil depends much more on your drainage than anything else. Assuming once you get the plastic up there is some sort of soil substance, that can be fixed without bringing in more unknown soil (that likely needs to be fixed). So if your soil drains from the house away from the house toward the street or other low spots off the property, then you don't need more soil. If the previous owner removed soil to bring in the rock, then you might need more soil. Drainage is the only important factor. If you bring in more soil have it done and leveled by a landscaper with a tractor and box blade. A bobcat or skid steer won't work nearly as well as a tractor.

Be sure the soil is perfectly prepped (level) before the sod is delivered. Sod sitting on pallets will either dry out or get a disease on the pallet. As for putting it down, YouTube is your friend. When you put the sod down, you need to have every piece butt up against the next piece so there are no holes or gaps. Use a machete or other large cutting blade to fit the edges or corners. Overlap the pieces of sod so they do not fit like a checkerboard. They should overlap from row to row like the slats of a wooden floor. Then when it's all down, roll it all with a rented, water fillable, roller. The idea is to push the bottom of the sod down to touch the top of the soil. Roots will not grow through the air to reach the soil. When you water the new sod, all you need is enough to keep the top of the soil moist - don't saturate it every day.

Need to know where you live.