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For the most part, this is completely normal. They drop a majority of the fruit in order to make sure that what stays on the tree gets the nutrition it needs. Here is an article for help. It is about lemons, but all citrus have the same general care and problems. The article can be found here: https://www.gardeningknowhow.com/edible/fruits/lemons/lemon-tree-fruit-drop.htm

Yes, you should replace the soil to the same level it was at before the soil washed away. Don't bury the roots deeper that they originally were, though, because this can damage the roots. You can also place mulch over the new soil to prevent more erosion, or even plant a shade-tolerant grass, ground cover, or moss around it.

This sounds like the tree is taking up too much nitrogen. You can offset this by adding nutrients high in potassium and phosphorus. Here is an article that will help: https://www.gardeningknowhow.com/edible/fruits/citrus/citrus-fruit-thick-rind.htm

Keep in mind that these trees will only survive down to 18 F, and will likely drop their fruit after a week or so at these temperatures. Here is an article that will give you more information about these trees: https://www.gardeningknowhow.com/edible/fruits/lemons/suckers-on-lemon-trees.htm

If the tree was grown from seed, it may not be old enough to produce flowers and fruit- citrus trees grown from seed vary in how fast they mature. If it was a grafted tree, it should be old enough now.

Check whether the tree is getting enough light and whether it's been fertilized appropriately. Kumquats especially need enough zinc.
https://www.gardeningknowhow.com/edible/fruits/citrus/fertilizing-citrus-trees-best-practices-for-citrus-fertilizing.htm

This article lists other possibilities:
https://www.gardeningknowhow.com/edible/fruits/kumquat/kumquat-not-flowering.htm

It would not contain viable seeds. It is only the hardened shell of a fallen fruit.

This is likely caused by a lack of phosphorus and potassium, or too much nitrogen in the soil.

You can counteract this by adding phosphorus and potassium, or by raising the pH of the soil a little with dolomitic lime.

This article will give you more information on the care of these: https://www.gardeningknowhow.com/edible/fruits/kumquat/growing-kumquat-trees.htm