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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

Roots of peach trees are not invasive, but they do need room to spread. Probably double the diameter of the foliage spread of the tree when mature will be room enough. The main thing to consider is light - your tree should get full sunlight all day for best fruit production. This article will tell you more: https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-care.htm

This is peach leaf curl. This article will help: https://www.gardeningknowhow.com/edible/fruits/peach/peach-leaf-curl.htm

The fruit may not be any good, as the fungus that causes the disease can also infect the flowers and fruit. However, often the diseased leaves fall off, and a new growth comes in, which may include new flowers and fruit. Some people remove and discard the infected leaves. Plan to spray with appropriate fungicides in the fall from now on. This article will tell you more: https://www.gardeningknowhow.com/edible/fruits/peach/peach-leaf-curl.htm

If the droplets are kind of gummy in nature, it could be from peach borers, which can affect the fruit as well as the tree. Here is more information that will help determine if this is the problem and what you can do about it: https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-borer.htm

There are several things that may have happened.

One is that it has been a cold and long winter. The flower buds may have all been killed off by a late cold snap.

Second, sometimes fruit trees do skip years, especially, as you describe, they had a really heavy year the year before. It is thought that the tree may be recuperating and fails to produce blossoms while it is recuperating.

Third, there may be an imbalance in your soil. I would recommend having your soil tested. This article will have more information on that:
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/testing-soil.htm