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This can have a few different causes. When the soil is too dry, if it gets too cold in the area, or if it is suffering from a bacterial/fungal infection.

If the area receives enough water, then you can rule out this possibility. If there has been a recent cold snap then this is a very likely culprit. If it hasn't been chilly, then you can peg this as an infection of some sort. Iron sulfate will, both, help keep the pH low enough to its liking, and kill off any infection in the soil.

Here is an article with more information that will help: https://www.gardeningknowhow.com/edible/fruits/citrus/curling-citrus-leaves.htm

Your lime tree might survive- it can be hard to tell until springtime comes how well it's going to recover. First, protect it from any future frosts by moving it indoors or covering it if more cold weather is predicted. Next, don't prune the tree until spring. Wait to see if it starts to grow leaves, and at that point it will be easier to tell which parts are dead and which are alive, and you can prune accordingly. Also see these articles:

https://www.gardeningknowhow.com/ornamental/trees/tgen/tree-scratch-test.htm
https://www.gardeningknowhow.com/plant-problems/environmental/how-to-treat-frozen-plants.htm
https://www.gardeningknowhow.com/plant-problems/environmental/tips-for-saving-cold-damaged-plants.htm

The best time to transplant lime trees or other citrus trees is when they are dormant- not putting out new leaves or fruit. This is usually in the winter.

Some small fruit drop is normal but it can also be caused by watering issues, temperatures fluctuating, pests, disease or improper pruning.
These articles will help you.

https://www.gardeningknowhow.com/edible/fruits/lime/is-lime-fruit-and-lime-blossoms-falling-off-tree-normal.htm
https://www.gardeningknowhow.com/edible/fruits/lime/problems-lime-trees.htm

This could be due to many things. Cooler weather can do this, as well as really hot weather. It could be not getting enough water as well.

Another thing that could be happening is that it is getting too much nitrogen, and not enough phosphorus, and potassium. There are citrus fertilizers that are designed for these trees. I would recommend using this once or twice per year.

This article will give you more information on lime care: https://www.gardeningknowhow.com/edible/fruits/lime/lime-tree-tips-care-of-lime-trees.htm

This will, almost always, have two causes. Lack of humidity, and lack of light. If you want this to fruit indoors, you will have to have supplemental light. These will require at least 200 watts per tree. I recommend, and use myself, a good LED panel. These will give them good light, and keep the cost to light it down.

It is also helpful to know that you will want to have, both, dolomitic lime, and iron sulfate handy. You will want to apply these once every few months. This will solve many problems before they happen.