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Your tree may be infested with Aphids.
The sticky substance is honeydew.
https://www.gardeningknowhow.com/plant-problems/pests/insects/aphids-tree-disease.htm

They grow best in a well draining soil that dries slightly between waterings.

https://www.gardeningknowhow.com/edible/fruits/lime/care-of-your-kaffir-lime-tree.htm

If there are no other symptoms then the most common issue here is iron deficiency. Feed with iron sulfate and the new growth will be unaffected. The old growth will not lose the spots, but they won't spread. Here is a link that will help: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/iron-for-plants.htm

It will likely recover. Depending on how cold it can get in your area, sometimes this can help protect a tree. The ice will remain closer to right at freezing, if the temperatures are way below.

You can prune any growth that is dead. To check, bend the stem. It should feel bendable. If it breaks, then it is dying or dead.

Yes. These are very alkaline in nature, and will quickly kill off plants. Try and add iron sulfate to counteract some of that, but if it continues, it will likely die.

I would try a root stimulating fertilizer like root n grow. This fertilizer is a 4-10-3. Any fertilizer with a higher middle number (phosporus) in it's npk ratio, will help promote root growth. Bone meal might also help.

http://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/bone-meal-fertilizer.htm