What's your question? Ask

Deer will normally eat anything when hungry enough, so there's really no guarantee that they won't bother your weeping cherry. That said, there are some things you can do, including adding 'deer resistant' plants. Here is more information: https://www.gardeningknowhow.com/plant-problems/pests/deer/deer-repellents.htm, https://www.gardeningknowhow.com/plant-problems/pests/deer/how-to-protect-trees-from-deer.htm, https://www.gardeningknowhow.com/plant-problems/pests/deer/deer-resistant-plants.htm

Trees normally start sending up more suckers when they are under more stress than usual. You can cut the suckers off, but I would not cut the roots out, as this will cause the tree more stress and possibly kill it. The best way to reduce the suckers is to figure out what is causing stress to the tree and reduce this stress. It will put out fewer suckers.

Mostly, growers start new weeping cherry trees by grafting cuttings on to hardy root stocks. Some success with rooting softwood cuttings under intermittent mist has been noted. No reason you can't try - take cuttings from the ends of the branches, 2 leaves and 3 or 4 nodes, dust with rooting hormone, and place in rooting medium, and cover to conserve moisture. Use a lot of cuttings, if you're lucky, a few will root. Here's an article with info about rooting cuttings: https://www.gardeningknowhow.com/ornamental/trees/tgen/how-to-root-cuttings-from-various-shrubs-bushes-and-trees.htm

Weeping cherry trees are related to sweet and sour eating cherries, but they re grown for their ornamental flowers and foliage, rather than fruit. While they do produce small fruits, humans consider them inedible, although many birds relish them. This article tells you more about weeping cherry: https://www.gardeningknowhow.com/ornamental/trees/weeping-cherry/weeping-cherry-growing-tips.htm

Is it soot-like in nature? If so, this is sooty mold fungus, usually brought about by insects, like aphids. You can treat the tree with neem oil, which will take care of whatever pest may be attacking the tree as well as the fungus. Here is more information: https://www.gardeningknowhow.com/plant-problems/disease/how-to-get-rid-of-sooty-mold.htm, https://www.gardeningknowhow.com/problems/pests/neem-oil-uses.htm

If the tree isn't girdled, it will likely recover. I would cover the lower trunk with a protective paper or tubing to protect it from further damage. This article may also help: https://www.gardeningknowhow.com/plant-problems/pests/how-to-keep-rabbits-out-of-gardens.htm