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If the trees are young and healthy it should be OK to separate them. If the tree is mature it may not survive the separation. Do it when the tree is dormant to help prevent disease and insect injury.

The leaves look spotty, like they have a fungal infection. The best thing you can do is prevent it next year. Clean up all the fallen leaves and dispose of them in the trash. Water from below; avoid wetting the leaves. The soil looks pretty dry, so you may want to increase the water.

The shrub prefers part shade. So if you have it in full shade you may want to move it.

https://landscapeplants.oregonstate.edu/plants/betula-nana

There are several birch types that will grow in your zone 8, including the river birch. Here is more:

https://www.gardeningknowhow.com/ornamental/trees/birch/10-types-of-birch.htm