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This tree is old- Very old. Likely, at the end of its life cycle, old.

Cutting the tree in half may or may not kill it. Either way, I would have a professional arborist to help with this. It could be a very dangerous task to take on without proper equipment and knowledge.

https://www.gardeningknowhow.com/garden-how-to/design/professional/when-to-call-tree-cutting-professionals.htm

This is a tough one, unfortunately. This article will help you to get started, though:

https://www.gardeningknowhow.com/ornamental/trees/tgen/tree-sucker-removal-and-tree-sucker-control.htm

These articles may help:

https://www.gardeningknowhow.com/houseplants/bonsai/bonsai-fruit-tree-care.htm

https://www.gardeningknowhow.com/houseplants/bonsai/bonsai-basics-starting-your-own-bonsai.htm

Sweet cherry trees, being susceptible to infection, will not be pruned during normal dormancy. With this being said, late summer is when to do so with sweet cherries. This is a bit late in the year, but you should still have enough time to prune before any chance of frost sets in.

Here is an article that will help you to prune cherry trees:

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

It would depend on how far it has been cut back. Weeping cherries are normally grafted and if it was cut under the graft, you should pull it out. You may get suckers from the root system but they will grow into an ornamental, multi-stemmed shrub within the same family and will not mature to the weeping shape.

I am not a current grower but here are a few personal observations.... Sour cherry trees are much more cold hardy and less prone to disease. Sweet cherry trees are more prone to infestations by insects, such as the cherry fruit worm and cherry aphids. Apply a horticultural oil during the dormant season and a fungicide in the spring to ward off diseases. Cover trees with netting while the fruit ripens... to thwart the birds.A sweet cherry tree might not bear fruit for at least five years. Unlike sour cherry trees, sweet cherries are not self-pollinating. You must plant two different cultivars to get adequate pollination and harvest. Yes, it can be another dwarf cultivar with a similar blooming time. Plant sweet cherry trees within 100 feet of each other, choose trees with similar bloom times (for pollination purposes) and avoid using pesticides in the garden, which might kill the honeybees needed for pollination. Sweet cherries are generally larger, and a lighter red and their pulp is thicker/firmer, a cream-yellow color on the inside. The fruit of sweet cherry trees usually looks more heart-shaped versus more rounder for sour cherries.