Will a tree grow back after stump grinding?
Short answer: the stump will not grow back once it is ground below grade. What can come back is shoots from the roots, and only for certain species. In central Iowa that usually means willow, cottonwood, elm and tree of heaven.
What grinding actually removes
The stump is the tree's storage organ. Grinding destroys it, and the roots lose the thing that was feeding them. The roots stop growing, and over several years they decay. No stump, no regrowth from the stump itself.
Roots are a different story. Some species have root systems that can send up new shoots from stored energy, especially when the tree was cut while it was still healthy and the roots were full of carbohydrates.
The species that sucker
Willow and cottonwood
The most likely to come back, and the most persistent. Both grow near water in the metro, both send shoots from roots 10 or 20 feet from the stump, and both can keep doing it for a couple of seasons. Grinding 12 inches deep reduces it. Mowing the shoots a few times usually finishes it.
Elm and tree of heaven
Both spread by root and both are common in fence lines and alleys. Tree of heaven is the worst of the two because the shoots come up fast and in numbers. If you have one, tell the crew before the job rather than after.
Maple, ash, oak, walnut
These rarely sucker from the roots after grinding. You may see a few shoots right at the edge of the hole in the first season, and they are easy to knock off with a mower.
What to do about it
Three options, in order of effort. Grind deeper, which is the cheapest thing to do at the time of the job. Mow the shoots, which works for light pressure and costs nothing. Or apply a stump killer to fresh shoots, which is a homeowner decision and something to read the label on carefully, especially near a garden or a waterway.
What does not work: covering the stump with soil, dumping salt on it, or drilling holes and filling them with anything. Those are old tricks that either fail or damage the soil around them.
Will a new tree grow where the old one was?
Not from the old stump. But the roots of the removed tree can release compounds that suppress new planting for a while, and the soil is disturbed. If you are replanting, grind deeper, plant slightly off center, and expect to water more in the first two years.