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It is best to only take off what you need. Leave as much as you possibly can to ensure its survival. You can definitely prop the tree back up with something whether it be a board, or tie it up with soft rope. I believe this tree will survive if given the proper care.

Yes that would help. Any fertilizer high in potassium and phosphorus will do it some good. Alternatively, the soil pH could be too low. In this case you can add dolomitic lime to the area. This will help buffer it back. Her is an article for more information on the tree:https://www.gardeningknowhow.com/ornamental/trees/silk-floss/floss-silk-tree-planting.htm

It should have already begun to show signs of life by now. It should only be watered when the soil is almost completely dry. These are somewhat drought tolerant, and can get fungal infections in wet soil pretty easily.

You can test to see if it is still alive by breaking just the tip off of a branch. If it is brittle, and brown inside, then go down until you find living tissue. If there is none, and the whole tree is brittle, then it is likely dead.

I would treat the area with wettable sulfur and dolomitic lime. This will kill off any infection in the soil, as well as recondition the pH back to a more suitable condition.

After this, you can replant, or wait for the tree to come back if it is still alive.

This is likely a pest issue; Scale insects are common on a Silk Floss Tree.

The substance that is being dropped by the tree is called Honeydew and is the bugs secretions.

Neem Oil is a good treatment.

https://www.gardeningknowhow.com/ornamental/trees/silk-floss/floss-silk-tree-planting.htm
https://www.gardeningknowhow.com/plant-problems/pests/insects/control-plant-scale.htm
https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm

It could be an insect, but it could also be a fungal infection. Do you have any more evidence of damage or anything?

It can be best to consult with your local extension service, as they can best help you with problems local to your area.

This link will help you to find the closest one to you: https://www.gardeningknowhow.com/extension-search

If this is a fresh transplant, then it is definitely going through transplant shock. It is not advised to plant in the heat, and it can result in the death of the plant.

Even though it would seem logical to water constantly in the heat, these trees are somewhat drought tolerant once they establish. There is a fine balance between keeping a young transplant watered well, and overwatered.

It is best to plant these when it is cool, but there is still a good chance that this will recover once the climate is right.

This article will help you with the care of these trees: https://www.gardeningknowhow.com/ornamental/trees/mimosa/chocolate-mimosa-trees.htm