When it's raining, check where the water is going. Often downspouts clog and water leaks from the joints, down the wall and finally into the basement. Also, check that the grading is moving the water away from the foundation. If the sump pump is working, be sure the water is moving away from the property--whether it's to daylight further out or to a sewer/storm drain. The occasional water you get in the basement is likely a result of poor external drainage on the foundation wall--no plastic membrane or similar stout barrier against the wall combined with slow draining soils. As a rule, tract builders simply backfill with whatever they've excavated. Instead of caulking around the water supply, a more permanent remedy would be a hydraulic cement based material. Frankly, the bit of water is nowhere as disturbing as that horrendous diaper insulation that is guaranteed to lead to problems.

If you're not going to water in the summer, then anything but St Aug will be better than St Aug. It will die if not watered once a month or so. If the zoysia does not do well in the shade, then St Aug is the only option. So then, in prep for sod, no need to rototill or aerate. Just rake or scrape off the surface dead growth. Then lay the sod. Make sure they leave no gaps between sod pieces. The sod pieces should be staggered meaning that it should not look like a checkerboard. They should roll the sod down to ensure good contact between the roots and the soil. Then start watering briefly (5-10 minutes), 3x per day (breakfast, lunch, and dinner times), for the next 2 weeks. Then check a few pieces to see if they have knit to the soil. Try lifting a corner. By then the roots should be biting into the soil. Then you can back off on watering until you are watering deeply (1 inch) and infrequently (once per week max during the hottest heat of summer). That watering schedule may be much different from what your neighbors say and all your local advice, but their daily watering schedule does not develop a great root system. Daily watering also encourages every weed seed to sprout, and you don't want that. So as soon as the sod knits down, start backing off on the watering frequency. Are you interested in an inexpensive organic fertilizing routine. Here is a picture of a zoysia lawn with a handful of alfalfa pellets (rabbit chow) applied a month earlier... You can see the improved color, density and growth. Alfalfa is usually applied at 15-20 pounds per 1,000 square feet. You can buy it at any feed store in 50-pound bags. You and your daughter might appreciate organic.

My experience (long ago) with "bagged topsoil" is that it is mostly cr*p. The Scotts name might, however, assure some quality. "So called" topsoil is likely to be an accurate description. I agree that you need to just mix compost into the topsoil you already have. I don't think peat moss will really serve any purpose, unless you're trying to lighten up clay. It's pretty pricey as well.

If it looks that bad, I don't think I would want an exchange, I would want my money back. Read the warranty and just stick to your guns. Don't accept another product that may get them through the warranty period but will ultimately fail too. If they insist, then tell them you want the warranty restarted. Get everything in writing before it is reordered.