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You could try this home made insect spray; pepper and garlic spray 1/2 cup of fresh hot peppers 2 large cloves of garlic 2 cups of water 1 Tbsp. vegetable oil. Liquefy ingredients in a blender and strain the mixture through a sieve lined with a handkerchief. Transfer into a spray bottle and label. Spray the upper and lower leaf surfaces of plants that show insect damage. This spray contains enough hot volatile oils to burn the chewing and sucking mouth parts of insects, but wear rubber gloves and eye protection as this can burn your skin and eyes, hope this fix's your problem

This article will help with that: https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-care.htm

Sounds like peach borer. This article will help:
https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-borer.htm

It is likely an animal, possibly birds. This article will help:
https://www.gardeningknowhow.com/edible/fruits/fegen/protect-fruit-tree-birds.htm

Yes, you can. This article will help: https://www.gardeningknowhow.com/edible/fruits/peach/planting-peach-seeds.htm

If you pruned while it was in fruit, it may have stressed the tree and made it more susceptible to disease or pests. In the future, prune your peach tree after you have harvested the fruit. This article will help you with pruning: http://wilkes.ces.ncsu.edu/files/library/97/How%20to%20Prune%20Peach%20Trees-4.pdf

I would recommend treating the tree with both a fungicide and a pesticide, to take care of any possible issues. Neem oil is both a fungicide and a pesticide. It is often sold as "Fruit Tree Spray" and would be good for treating your tree.