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The weeping cherry is hardy to minus 10 degrees, and will survive in your area naturally. If you still have concern, move it to a south facing wall or mulch it with hay. Here is an article that will help: https://www.gardeningknowhow.com/ornamental/trees/weeping-cherry/weeping-cherry-growing-tips.htm

Yes, this is quite common. Transplanting is rough on any plant. This stress will cause the flowers to drop. Just make sure to keep the tree watered, but not sopping wet. Don't let it dry out until it is established, and has good new growth.

Here is an article for more information on the care of these trees: https://www.gardeningknowhow.com/ornamental/trees/weeping-cherry/weeping-cherry-growing-tips.htm

The branches won't sprout directly from the ends where you pruned them, but they will put out more side branches from buds closer to the trunk. Proper pruning can correct your tree's shape. It may be best to contact an arborist or other professional for help. This article will help you learn about weeping cherry tree pruning:

https://www.gardeningknowhow.com/ornamental/trees/weeping-cherry/pruning-weeping-cherry.htm

Also see these articles about tree pruning trechniques:

http://www.extension.umn.edu/garden/yard-garden/trees-shrubs/pruning-trees-shrubs/
https://extension.psu.edu/natural-resources/forests/urban-community/publications/pruning-landscape-trees

Many cherries do develop a brown bark. Some even peel! Unless it is affecting the leaves and flowers, I would not be worried. This article will give you more information on the care of these: https://www.gardeningknowhow.com/ornamental/trees/weeping-cherry/weeping-cherry-growing-tips.htm

This would be ok right now, but I would not wait much longer. You can trim as high as you would like, but most prefer to have them just above the ground.

Without a soil test, you'll never know for sure. Some advise a slow-release all-purpose, balanced fertilizer in spring when leaves are emerging. Balanced means the NPK numbers are similar, 10-10-10 or 8-10-8 for instance. Nitrogen leaches through the soil every season but phosphorus and potassium build up and can reach toxic levels. Some soils are naturally high in phosphorus and adding more creates problems. You can look to your state's cooperative extension service for general info on soils in your area. Some gardeners choose to loosely rake in 2 inches of compost every year as an alternative to fertilizer. If your tree doesn't look healthy, your first step is a soil test.