In our road we have a crab apple tree that has fruits but there have never been any small new trees rooting from the crab apples. Once, years ago, there were 3 crab apple trees within 100 yards but 2 of these were destroyed by some ignorant house owners. However in my 83 years of living within 50 yards of these three trees I have never seen any young new seedling trees. Some of the apples must have fallen onto the surrounding soil that is never disturbed. Will an apple tree grow if I deliberately plant some of the fruit from this one remaining tree?
Not if you just plant a whole fruit, no, very unlikely. There is a way to do it, but you need to extract the seeds from the fruit in autumn by cutting it open, lifting the seeds out carefully, then removing any remaining pulp by flushing with water. After drying, they need cold stratification for up to 120 days - a description of how to do all this is shown here http://homeguides.sfgate.com/propagating-crabapple-seeds-27860.html, but you should bear in mind that Malus varieties do not come true from seed. The named varieties you buy are vegetatively propagated to ensure they are the right variety - from seed, you don't know what you'll get, other than it'll be a crabapple of some sort.
I don't really know for crab apples, but I can tell you my experience with germinating regular apples (not necessarily on purpose).
Anyway, I've found that if the ground is bare (has no grass, weeds, or anything on it), and if I squash the fallen apples on the ground, the next year I tend to get a lot of apple tree seedlings growing. If there's a lawn or anything like that, I wouldn't count on it (we had a lawn for years with few if any trees ever germinating in it, but now that we don't have a lawn there anymore, they germinate like weeds, especially Northern Catalpa, Lombardy Poplar, and Siberian Elm, but also apple and maybe other stuff). Trees like to germinate in bare soil, in my experience (not just apple trees). I don't know that I ever had seedlings from apples that I didn't squish (but it's possible; we usually picked most of the non-squished ones up).
Anyway, I didn't plant the apples at all. I just squished them flat on compact, almost clay-type soil.
We do get cold winters (which may or may not be necessary for germination).
I wouldn't be surprised if grass needs to be absent for a number of years. It seems like the trees didn't start germinating in large numbers until we went a few years with no lawn.
If you're wondering why I would squish apples (and I'm not recommending people squish them), it's so they would decompose faster, and so the nutrients would go back into the soil. We've had a lot of apples drop before they were ripe, probably for various reasons (drought, no mulch, old tree, perhaps a lack of thinning, severe, chronic spider mite infestation, maybe disease, and maybe other stuff).
I don't believe I've ever had seedlings grow from planting entire apples. I only recall burying apples once (but not to try to germinate them); none sprouted. Nevertheless, I probably buried them too deeply for germination, and in soil that dries out very easily (not much grows in there, other than catnip in the spring).
Your crab apples probably don't have the genetics to breed true-to-type from seed, and they'll likely also be cross-pollinated, if they're like regular apples (if other apple or crab apple trees are near). But, you can certainly grow trees from seed, if they're like regular apples (I know they can pollinate regular apples).
Yes. You can learn a lot from this extensively reworked and thought-out answer to a similar question about roses: How do I grow roses from seed?
Fruits of all kinds are attractive to wildlife, and are presumably picked up and eaten or stored in underground dens where the seeds never see the light of day again, or immediately eaten and deposited later in a remote bank. Seeds may well have built-in dormancy which ensures that they do not germinate for years to spread out the risk. But sometimes you can wait too long.
Just burying the fruit or seeds probably won't give you the hoped-for result. It's a matter of making it a special project and putting in lots of TLC to make sure the trees survive.
If you think about the reason why fruit exist, it is to employ animals to disperse the seeds far away from the parent tree. So if this works (which we can assume it does based on the principles of evolution) one might expect that not many of the seeds will remain near the parent for the requisite amount of time (the cold of winter, the warm of spring) to induce germination. Those seeds that are dropped relatively nearby will face a lot of competition: shading from other plants, predation, gardening activities of humans, etc.
The previous comment about apples not coming true to seed is valid, except what this usually means is that cultivars of apple result in something more like crab apples, which is not a problem in your case. If you plant a bunch of seeds and keep them the cold over the winter, you should get some saplings in the following year.
The short answer is yes. There's a lot of nuance to this though. If you select a fruit from the nearby tree that's ripe enough to have viable seeds (black or brown not white) AND they are harvested correctly from within the apple AND they go through the proper stratification process AND they get the right conditions to sprout then you're well on your way to maybe an apple tree.
I say maybe because the fruit you harvest the seeds from will not be the fruit your tree would grow up to be. Since nearly all apples require cross-pollination you'll end up with a hybrid fruit from the parent tree as well as the pollen from the pollinating tree. Could be something really awesome and you just created a new cultivar. Likely, however, it will be some sort of crabapple-type tree. Especially since you mentioned that you have crabapples in your area.