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Do you keep the lawn watered? Ants tend to build in dry sites. These articles should help:

https://www.gardeningknowhow.com/plant-problems/pests/insects/how-to-control-ants-in-lawns.htm

https://www.gardeningknowhow.com/plant-problems/pests/insects/get-rid-of-ants.htm

I wouldn't use Dawn if you are making insecticidal soap. (Use one without a degreaser.) It's also recommended to buy insecticidal soap for a safer product instead of making it. Homemade ones are more likely to burn the plant.

https://www.gardeningknowhow.com/plant-problems/pests/pesticides/homemade-soap-spray.htm

https://hgic.clemson.edu/factsheet/insecticidal-soaps-for-garden-pest-control/

Neem oil, like insecticidal soap, has to be reapplied. Make sure you are following label instructions and are spraying the top and the underside of leaves.

https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm

Here is comprehensive info on controlling bugs. If you decide to go with more potent pesticides, applying in early morning or late evening will avoid most pollinators.

https://extension.tennessee.edu/publications/documents/PB595.pdf

https://ag.umass.edu/home-lawn-garden/fact-sheets/insect-management-in-home-vegetable-garden

It is hard to say which species, but it is a type of caterpillar. They can be quite destructive. Neem oil is a good way to get rid of them, but bad infestations can require other methods.

bacillus thuringiensis or BTK is one of my favorite methods. This microbe is good at getting rod of caterpillars while being harmless to you or your plants.

https://www.gardeningknowhow.com/plant-problems/pests/pesticides/using-bacillus-thuringiensis.htm

This article will offer more information:

https://www.gardeningknowhow.com/plant-problems/pests/insects/prevent-caterpillars.htm

It is. What you can do is to destroy its egg sacks before they emerge. This, paired with having reptiles or larger birds like chickens, can reduce the invasive population quite well.

It will be best to look on the container, since the manufacturer will have a specific date. This can be a guideline for a rough estimate of how long it will be effective.

As far as disposing of permethrin- Check the label or packaging of the spray for any specific disposal instructions provided by the manufacturer. They may have specific guidelines for proper disposal. If there are no specific instructions, you can dispose of the spray can in your regular household trash. Make sure the can is completely empty before disposing of it. Spray it in a well ventilated area, like outside, then it will be safe to dispose of the empty can.

Yes, it is recommended to scrape off the dead grass and reseed. The timing depends on which grass you choose.

https://www.gardeningknowhow.com/lawn-care/lgen/lawn-seeding-tips.htm

Getting rid of them is best done, naturally. Neem Oil, milky spore, and beneficial nematodes are helpful. There are chemical options too.

This article will help:

https://www.gardeningknowhow.com/plant-problems/pests/insects/get-rid-of-grub-worms.htm