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These are fungal fruiting bodies/mushrooms growing on what appears to be dying grass. They are a seasonal spore dispersal body and will dry up and disappear in time. Or you can use a lawn mower or broom to knock them down.
If you improve the health and density of the grass with re-seeding, fertilization and water management the fungus will likely be suppressed.

It's hard to tell from the photos how large the holes are, and you don't state the diameter, but the following guide may be of help in determining which critter you have. I suspect ground squirrels or maybe voles if the holes are small.

http://icwdm.org/Inspection/GroundHoles.aspx

They are not directly linked together, but anytime an insect draws sap, then it will likely draw more pests. This will compound the issue.

Insecticidal soap is not a very effective treatment for infestations. This is more for seeing a single insect, or maybe a small cluster and treating them there. This will require some pesticides, whether you choose natural or organic.

This is our collection of organic articles which will offer many ways to control pests quite effectively. any of these recipes will be using ingredients that you likely already have around the home.

https://www.gardeningknowhow.com/special/organic

There are many possible reasons; here is an article that may help you to determine the cause.

https://www.gardeningknowhow.com/lawn-care/lgen/reasons-for-dying-grass.htm

Moles, skunks and raccoons will dig and disturb grassy areas. Skunks and raccoons are looking for grubs and other foods. Sometimes, signs of digging mean there is a grub problem that should be treated in the spring with GrubEx or other appropriate pesticide. For now, walk the lawn to try to tamp down the disturbed areas and get roots back in touch with soil. Here is more information on lawn skunks, raccoons and moles and what to do about them: https://www.canr.msu.edu/news/reduce_lawn_and_garden_damage_caused_by_moles_skunks_and_raccoons

Manual weed control, pulling, digging is the best but requires physical work.
Chemical weed control with the "weed and feed" type of product you mention is possible. That will kill broadleaf weeds and not grasses, if you follow the label instructions and precautions.
If you have trees in the area be advised that some broadleaf weed killer chemicals can harm trees if the chemcials are taken up by absorbing roots in the area that is treated.