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That's hard to say without photos or description of the current condition. It is possible to fry new seedlings with excessive high Nitrogen chemicial fertilizer. If the grass is browned out and dead looking then you may have to start over. If it's green and growing then keep up the watering and hope for the best.

I see this as a test of our problem-solving skills. Why was the area bare? How was the ground prepared? Has herbicide been used in the area recently and which one. (Some stay active much longer than others.) Where you watering 3 times daily during hot weather? Tiny new roots can't support grass in full sun during mid-day August heat. Is it too shady for grass to succeed? Did you use a regular fertilizer instead of a starter fertilizer? "Adult strength" fertilizer might burn new roots. If all of this checks out and you have read the attached article, then try again in a month. As always with difficult situations, consider a soil test.
https://www.gardeningknowhow.com/lawn-care/lgen/lawn-seeding-tips.htm

Drought can lead to thin grass or outright dead areas. When weather cools, rake the lawn well and sow seed in thin areas. If areas have zero grass, prep the ground by loosening the soil, sowing seed and covering it with top soil, compost or straw. Keep the seed moist or it will die; this means watering at least twice a day without flooding the area. https://www.gardeningknowhow.com/lawn-care/lgen/lawn-seeding-tips.htm

Yes, you can plant these in early fall. These grasses are very hardy and will adapt to almost any condition.

You will divide them when they are mature. Just take a shovel and split parts of the outside of the clumps. You can plant these elsewhere.

This article will help: https://www.gardeningknowhow.com/ornamental/foliage/feather-reed-grass/growing-feather-reed-grass.htm

The grass seed at your local garden center will be hardy in your area. If the area gets some afternoon shade, consider one of the no-mow grasses like Eco-Lawn. There are a number of options out there. Grass alternatives are numerous but I would need to know how much sun the area gets to give suggestions. Will there be foot traffic? Is the soil more sand or clay? Then there are native plants which require minimal care after some watering the first year. Native grasses come in 2 basic varieties; those that form clumps making a nice round shape at 2' tall to loose, spreading 6 ft tall non-clumpers. Perhaps a mix of cool season and warm season grasses. Native sedges are usually 1-2 ft. and require zero maintenance after established. Search for the online native plant nursery near you. One that I buy from is PrairieMoon.com

Grass should have about 6 weeks of growth before a killing frost. Sowing in early September is a popular time but spring is good too. It is too late for grass to form sufficient roots to make it through winter in MO. Sow seed as soon as you notice the grass starting to green up next spring. New grass needs as much time as possible to get established before brutal heat arrives. You may need to water the new lawn during dry or very hot spells to make sure it makes it through the summer. Here are details on growing grass in Missouri: https://ipm.missouri.edu/MEG/2014/5/Seeding-and-Fertilizing-Missouri-Lawns/