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

This article outlines some common plum diseases: https://www.gardeningknowhow.com/edible/fruits/plum/plum-tree-diseases.htm
If it is a rot disease, you need to get it identified so you can treat it. Please get in touch with the nearest botanical garden and the Master Gardener program; they can help you.
Greengages plums are a very special fruit. In doing some research, I found this very interesting article: http://www.nytimes.com/2004/09/01/dining/01GREE.html?pagewanted=print&position=&_r=0

You should be in the right growing zone for a Greengage Plum Tree.

Your trees needs a full sun location with a good loamy soil that is well draining.
Ph should be in the 6.0 to 7.0 range.
Though this is a self fertile tree you may want to add an additional pollinator to your garden.
Fertilize your tree with a balanced fertilizer. This should be applied annually and a good rule of thumb is 8 oz of 10-10-10 for every year of tree age.
Supplement with nitro-chalk sulphate of potash in February each year.
Every third year add superphosphate.
Mulch the tree with a good compost, but do not place it directly next to the trunk.

Green gage plums will soften somewhat after you've picked them, but they won't reach their peak flavor unless they're allowed to ripen on the tree. This article goes into depth about it:
http://www.nytimes.com/2004/09/01/dining/a-finicky-fruit-is-sweet-when-coddled.html

There is nothing wrong with this at all! I believe it would focus energy to where you want it. Go for it!

If it still has fruit on it, you wouldn't prune it because in established plants, you would only prune non productive wood.

https://www.gardeningknowhow.com/edible/fruits/plum/plum-tree-pruning.htm