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You are likely doing just fine with your tree care.

Plum trees will not began to bear fruit until they are 3 to 6 years of age.
Here is a link with more information.

https://www.gardeningknowhow.com/edible/fruits/plum/plum-trees-not-fruiting.htm

Here is an article about weed trees and some suggestions on controlling this issue.

https://www.gardeningknowhow.com/ornamental/trees/tgen/weed-trees.htm

Check to see if the branches have died off. Bend them and if they snap they have died.
You can also scratch the branches, if still alive, you will see green under the bark.
These links have more information and how to winterize your tree.

https://www.gardeningknowhow.com/edible/fruits/fegen/winterizing-fruit-trees.htm
https://www.gardeningknowhow.com/edible/fruits/plum/growing-plum-trees.htm

If you have plum moths (Grapholita funebrana), beginning to spray in the third week of June is recommended. Or, you can use pheromone traps to determine timing more precisely:
https://www.rhs.org.uk/advice/profile?PID=554

The tree bands are recommended for different species of moths, and they should be applied in the fall:
https://www.rhs.org.uk/advice/profile?pid=518

If you're not sure of the moth species you have, contacting a local extension agent will help you find out what is common in your area.

It sounds like the rootstock has begun to assert itself on the tree. Typically you would want to remove the rootstock branches, but if you are enjoying the fruit, you can leave them.

As to the variety, it would be hard to say as there are several used for grafting. Here is a list of some typical rootstocks that are used for plums:
https://www.orangepippintrees.co.uk/articles/rootstocks-for-plum-trees

It sounds like bacterial spot, which is safe to eat.

https://extension.missouri.edu/news/DisplayStory.aspx?N=449
https://www.gardeningknowhow.com/edible/fruits/plum/plum-tree-diseases.htm