Your garden will likely be successful no matter how you prepare the soil. You could put down some compost, and then dig it into the sod when you plant your tomatoes, making a hole for the plug. You will have to do some weeding; I don't think that all of the weed seeds in the soil will have been killed by your newspaper barrier. If you set out a variety of plants, some will thrive, and others will not. Some plants will be able to grow on top of the sod, in the compost you have provided, and others will not succeed. You could rent or borrow a rototiller, grind up the sod, and mix it with your soil and compost. But I think you will have some success with planting garden vegetables over the sod.

Not knowing anything about your soil, I would generally dig in some compost if converting the average lawn into a garden. See how it does, and plan to add more compost next year or in the fall, more or less depending on how the soil feels and how well the garden grows.

I'd follow your original plan of topping the newspaper with as much compost as the stones can contain (unless that's higher than 8 inches or so?). You won't be growing in pure compost because the plants' roots will penetrate the newspaper and soil, but you'll have enough to smother and fully decompose the sod. Then after this season you can decide whether to dig/till or to wait and let the earthworms do it for you (slooowly).

I've done this for several beds and really enjoyed the results. My garden has clayey (might actually be silty according to the USGS) soil and ample moisture, so your results might vary in different conditions.

If you search this forum for posts by member 'jonhughes' (hope I got that right) he had a series of pictures showing a bed constructed this way with cardboard smothering layer.

~emmers

Where in the United States are you? That would be helpful to know since that provides some information about climate, rainfall, soil type (for some), average annual temperatures, etc.
What is the soil like? Clay? Sand? Silt? or a combination?
Whether the newspaper will kill the grass depends on what grass species was there. Newspaper will kill off wimpy grass species like Blue Grass, Perennial Rye, the Fescues, but not weedy grasses like Quack and Johnson grass. Although the top growth of those will die the roots will only be barely affected.
Add enough compost, or other forms of organic matter, to provide about 6 to 8 percent of soil total.
Have a good reliable soil test done for soil pH and levels of Phosphorus, Potash, Calcium, and Magnesium and look for balance.