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Yes, this is possible for them to cross pollinate. You would not notice anything unless you planted the seeds from the fruit. They would, then, express traits from both parents.

This is likely a plum canker. Most years it will not affect much. Some years, when it is vigilant, it will wreak havoc on the tree. There is no cure for this, but you can treat it every year in spring with dolomitic lime and wettable sulfur.

Ah... the Codling moth... This article will help: https://www.gardeningknowhow.com/plant-problems/pests/insects/controlling-codling-moths.htm

If you can include a photo, then it will help me to identify what is going on. In the meantime, this article will give you more information on the care of these: https://www.gardeningknowhow.com/edible/fruits/plum/growing-plum-trees.htm

This has two likely causes. The first would be underfertilization. I would make sure to apply a fruit tree fertilizer once per year, or a nitrogen based in spring, and switch to a high phosphorus and potassium fertilizer once the fruit start to form.

The second would be a fungal infection. This is common during wet seasons. Wettable sulfur, and dolomitic lime should be a once a year application, just to be sure, anyway. Add these if you, also, suspect a fungal infection.

This article will give you more information on the care of plum trees: https://www.gardeningknowhow.com/edible/fruits/plum/growing-plum-trees.htm

Do you know the exact variety of plum that it is? Can you include a photo? What is your USDA zone? Do you know the age of the tree?

There are many things that could cause this issue. This article will help in the meantime: https://www.gardeningknowhow.com/edible/fruits/plum/plum-trees-not-fruiting.htm