What has been feeding on my lemons? It is an extraordinary sight to see the lemons still attached to the tree with some or no white pith, exposed to the air, and there is lots of debris on the ground under the canopy. Have you ever heard of such a thing happening?
Yes, this is an increasingly common occurrence with lemons along with other types of edibles and non-edibles. Later this summer, tomatoes will be victimized.
It's improbable that a bug or insect is the cause of the problem as their mouthparts are not large enough to due to this type of damage. Also, it would take a large army of them, diligently at work twenty-four seven, to skin a lemon. It's much more likely that the culprit is a nocturnal rodent or larger animal. Raccoons, possums, or maybe squirrels come to mind; however, my number one suspect is roof rats.
Roof Rats are a different species from sewer rats. They can nest in structures such as the attic and crawl spaces but are more likely to nest outdoors in dense, protect them from predators such as feral cats.
They're very mobile, so they are maybe nesting elsewhere. The lemons are not a regular part of the diet; instead, they use them as a water source. Their likely to do more damage during a drought as their usual water site dry up. My New Guineas Impatiens were destroyed in 2015 by the roof rats devouring the fleshy stems.
Unfortunately, there isn't one ideal solution. Netting the lemon tree/bush isn't viable because of its fruit size and year-round maturity. Poison Baits are not advisable unless there is some bait station that would protect the domestic animal, aka dogs and cats, from it. You might trim the foliage back, so the access from some structures is reduced along with trimming up the lower branches.
Adopting a hungry cat to roam at night is not a bad idea; otherwise, you'll need to contact a commercial exterminator for a second opinion.
Our 20 year old Fuji apple tree always produces lots of sprouts around the base of the trunk. This year I dug one with lots of roots out and potted it. Will it grow and produce the same apple as the mother tree?
It's pretty common for apples to produce suckers at the base of the tree. The suckers are from the rootstock that the desirable variety was budded to. The chances it's a clone of the Fuji Apple is very, very, very, slim. Since you already transplanted it, I'd wait and see what happens but lower your expectations.