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I would have the soil tested to make sure that there is not a phosphorus deficiency in the soil. The trees need phosphorous to bloom and produce fruit. You may also have a pollination issue. Attracting pollinators will help. These articles will help: https://www.gardeningknowhow.com/garden-how-to/beneficial/insect-pollination-process.htm, https://www.gardeningknowhow.com/garden-how-to/beneficial/creating-a-pollinator-garden.htm, https://www.gardeningknowhow.com/garden-how-to/beneficial/attracting-bees.htm

It might be a nitrogen deficiency, but it would have to be pretty severe to cause that kind of damage. Root rot would be another possibility, but that would be unusual for this kind of tree. I would also check for pests. They will normally be found on the underside of the leaves or in joints where the stems and branches meet.

Well, you need to do some cleanup pruning. Prune out what is dead and any weak branches, especially if they are crossing other branches. Once the clean up pruning is done, if you have any fruit left on this tree this year, you need to thin it out. This article will help with that: https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-thinning.htm In the future, you will want to consider spraying the tree. This article has more on that: https://extension.missouri.edu/p/G6010-7

It sounds like galls or a fungus of some type, possible even black knot. A fungicide may help. This article will help determine if black knot is infecting your tree: https://www.gardeningknowhow.com/edible/fruits/plum/black-knot-tree-disease.htm

You can go ahead and plant the tree but make sure the planting hole is the same depth as the pot it's currently in and flush with the soil level to prevent pooling of water, which can lead to rot. Also, be sure to keep it well watered, especially if the tree is already trying to put out more fruit. These articles may help as well: https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-care.htm, https://www.gardeningknowhow.com/ornamental/trees/tgen/watering-newly-planted-tree.htm

Yes, it is likely birds. They are notorious for this. This article will help: https://www.gardeningknowhow.com/edible/fruits/fegen/protect-fruit-tree-birds.htm