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It sounds like the tree may have a fungus. Some fungus that get into the fruit can be treated, while others cannot. Your best bet would be to treat the tree with a fungicide. It may stop the fungus from spreading. Even if it does not, you can stop the fungus from returning next year by making sure that all of the leaves in the fall are properly disposed of.

This is sooty mold. It happens when the plant gets a pest infestation, likely aphids. I would recommend using neem oil, as this will get rid of the sooty mold fungus and kill the pests. Here is some articles that will help you:
https://www.gardeningknowhow.com/plant-problems/disease/how-to-get-rid-of-sooty-mold.htm
https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm
https://www.gardeningknowhow.com/edible/fruits/plum/control-leaf-curl-plum-aphids.htm

This article can help with reasons for non-fruiting: https://www.gardeningknowhow.com/edible/fruits/plum/plum-trees-not-fruiting.htm

I think it may definitely be the fertilizer from your lawn, which normally contains lots of nitrogen - this would result in lots of lush foliage growth with little to no blooms or fruit. You can usually offset the nitrogen by adding a phosphorus rich fertilizer, or bone meal, to the soil around your tree. Here is more information: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/phosphorus-plant-growth.htm

Hopefully this article will help: https://www.gardeningknowhow.com/edible/fruits/plum/plum-tree-diseases.htm

If the streaking is nearest the leaf veins, it might be a virus. I would suggest taking a sample to your local agricultural extension office for identification and recommendations for treatment.

It sounds like brown rot. This article should help: https://www.gardeningknowhow.com/edible/fruits/plum/plum-tree-diseases.htm

It sounds like the plant may have a fungus. Some fungus that get into the fruit can be treated, while others cannot. Your best bet would be to treat the plant with a fungicide. It may stop the fungus from spreading. Even if it does not, you can stop the fungus from returning next year by making sure that all of the leaves in the fall are properly disposed of.