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This is caused by peach borers. Treating with neem oil will help with getting rid of them. This article will have more information:
https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-borer.htm

There are many reasons for the surface roots, one of which could have been that the tree itself was initially planted too shallow. In addition, there may not be enough topsoil. As for the oozing sap, this could either be attributed to changes in temperature or a problem with peach borers.

The rain is likely the cause. Too much water in the soil can start to suffocate the roots. If you can, try to improve the drainage. Otherwise, you will have to wait for the rain to back off a bit so that the soil improves.

I would remove them. This will allow the tree to focus on growing roots and becoming established. There is a high likelihood that the tree would abort the fruit anyway, but the sooner you remove them, the sooner it can focus on what it needs to in order to survive and do well in your yard.

It is a disease called peach leaf curl. Treating with a fungicide will help. Here is more information on it:
https://www.gardeningknowhow.com/edible/fruits/peach/peach-leaf-curl.htm

How often are you watering? Trees need significant amounts of water their first year. I would recommend increasing watering.