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This article should help: https://www.gardeningknowhow.com/edible/fruits/citrus/fertilizing-citrus-trees-best-practices-for-citrus-fertilizing.htm

I would suspect a raccoon is the culprit. They can climb trees and are very crafty with their paws so they are able to 'pick' fruit from plants. They are also quite wasteful and will eat only a bite or two of anything they get. They are nocturnal and rarely, if ever, come out during the day - only at night. This article may help with keeping them away: https://www.gardeningknowhow.com/plant-problems/pests/animals/getting-rid-raccoons.htm

This article should help (though it refers to oranges, it really applies to any of the citrus fruits): https://www.gardeningknowhow.com/edible/fruits/oranges/oranges-splitting.htm

Continue to fertilize as usual. This article will help with fertilizing requirements for citrus tres: https://www.gardeningknowhow.com/edible/fruits/citrus/fertilizing-citrus-trees-best-practices-for-citrus-fertilizing.htm

This article may help: https://www.gardeningknowhow.com/plant-problems/environmental/tips-for-saving-cold-damaged-plants.htm

This is likely a fungus. Are there any other symptoms on the tree, such as the leaves or the bark? This will better help us identify which one exactly is hurting your tree. In the meantime, I would treat the plant with a fungicide.