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Did the leaves turn yellow before the fell? If so, your tree may be suffering from iron deficiency. Adding iron to the soil and using a foliar spray may help. Unfortunately, if that's not the case, most mulberry diseases prove fatal.

There are sprays available that can reduce the tree's ability to produce fruit. They are sold as "fruit eliminator sprays". They are available at many nurseries and certainly online.

You need to prune and shape it to weep. Normally you should do pruning when the tree is dormant. Use sterile, sharp pruners and prune no more than 1/3 at a time.

It may actually be animals. Deer or other antlered animals will use trees to rub their antlers on. This causes scratches and damage to the bark.

It may also be a canker fungus. There are several that can affect the trees. This guide will help you determine which it might be: https://www.extension.iastate.edu/Publications/SUL11.pdf

https://www.gardeningknowhow.com/ornamental/trees/fruitless-mulberry/possible-causes-of-a-fruitless-mulberry-with-yellow-leaves.htm

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It is hard to say. Normally, no, it will not cause problems. But if the roots are already constricted by something else, like a foundation or if the pipes already have a crack in them, it could cause problems.