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This appears to be brown rot of plum trees. It is a fungal disease. Here is some information on prevention and treatment. https://www.gardeningknowhow.com/edible/fruits/plum/treating-brown-rot-in-plums.htm

These appear to be Plums.

This does sound like a case of a pest. I would place my bet on birds, though. Possibly squirrels. They can make their way through just about any barrier. they are very intelligent. You can plant citronella geranium, mint, and other strongly scented plants around to help with this.

Are the seeds that you received already pulled out of the stone? If not, then it will take a lot longer to germinate.

If you can manage to crack out the small, almond looking seed inside (please be careful and wear gloves, this is very very toxic. Do your best not to knick the seed inside in any way) and plant this, then it will germinate in a month or two, rather than about 6 months.

Leaving it inside the shell that it is likely in will cause this process to take a long time, as this shell has to break down before the seed inside can germinate.

There can be numerous reasons why a plum tree drops its fruit before it ripens: insects, disease, high winds, and stress ( too much water, too little water, lack of nutrients, poor drainage, weed competition, and excessive shade). But the most common reason -- especially when, as you say the tree is heavily laden -- is that the tree has produced more fruit than it can support. If this happens again, you'll want to thin the fruit.
When the fruit reaches about 3/4-inch in diameter, pick off extra plums so that the ones that remain are about 4 or 5 inches apart.
https://www.gardeningknowhow.com/edible/fruits/fegen/reasons-for-small-hard-fruit.htm
https://www.gardeningknowhow.com/edible/fruits/plum/plum-tree-diseases.htm
https://www.gardeningknowhow.com/edible/fruits/plum/plum-curculio.htm

This is completely normal! They need some dormancy to ensure proper fruit set the next year. I am seeing new growth that will become flowers soon. Those knotty looking growths are early blooms. After the flowers, will come leaves. If you want to ensure that it remains healthy, you can always fertilize with an all purpose slow release for fruit trees, dolomitic lime, and iron sulfate once per year.

Other than this, the tree looks completely normal. Here is an article that will give you more information on the tree's care: https://www.gardeningknowhow.com/edible/fruits/plum/growing-plum-trees.htm