There are two potential circumstances where a bee colony loses its queen.
The queen is old. She is not laying eggs well anymore, which means that many, most, or virtually all of her eggs are left unfertilized, resulting in drones (males) instead of worker bees. This scenario requires urgent attention, because the hive will not survive without a steady brood production, and once all of the queen's fertile eggs are gone, there can be no possibility of replacing her.
Worker bees are able to lay eggs--though few are aware of this, and it seldom occurs in the wild. A worker bee's eggs would all be drone (male) bees from infertile eggs because the worker is not sufficiently developed to have even the possibility of mating with a drone.
Queen bees mate once, and lay hundreds of thousands of eggs over the next as many as six or seven years before reaching the end of their productivity. They store the sperm in their body and are able to choose whether to lay a fertilized or unfertilized egg--at least, up until those stores are gone.
So when it becomes apparent within the hive that the queen is at the end of her useful life, it is urgent that she be replaced. The workers will take some of her last remaining fertile eggs and place them in the special queen cells, fed by extra royal jelly (all bees get some, but queens just get a bigger dose of it, and the hormones in the royal jelly help the queen to mature). Whether before or after the workers have done this I am not certain, but the worker bees themselves will turn traitor and, for the good of the hive, kill the aged queen to make way for a new queen.
The queen is lost to some unforeseen misfortune. If this occurs during the winter hibernation period, the colony will perish, having no possible recourse. But during the breeding season, worker bees can take emergency measures to produce a queen from the last eggs laid. If there are existing downward-hanging queen cells on standby, they will likely relocate those eggs to them. If not, I have seen a few queen cells positioned horizontally like all of the rest of the brood, but made larger to accommodate a queen's size. This may be what the Wikipedia article references.
Beekeepers do not usually appreciate the sight of queen cells, and tend to destroy them whenever they are discovered, for several reasons: 1) They do not wish to have rival queens develop that might fight with and kill their special hybrid queen (queens can be purchased from various breeds, including Carniolan, Italian, etc.) by which the beekeeper has sought to balance the temperament of the hive with the environmental conditions at the site; 2) beekeepers know that bees naturally will split, with a large part of the colony following a new queen to start a hive in a new location, if new queens emerge. This will set back the production of the original hive as many worker bees are lost to the emigration; and 3) queens which fight well (there can only be one queen in a hive, so queens will battle it out to the death) are not necessarily the best egg layers--therefore, if a queen must be replaced, the beekeeper would prefer to take charge of this.
Regarding the specific questions under the main one--and, again, these are answered according to my understanding:
Worker bees will move an egg that is not more than about three days old. Beyond this it might be impractical to move it. This is why action must be swift if there is a loss of the queen. As eggs progress through the larval stages, they increase in size within the cell. They are fed, cleaned, and generally pampered (climate controlled, etc.) throughout the process. But a large larva could not be easily moved.
I've never heard of a drone emerging from a queen cell, and believe the worker bees must be able to distinguish fertilized eggs from unfertilized eggs. That is the only point of significance to egg selection. Any fertilized egg could become a queen.
In order to become queens, the larvae must consume adequate quantities of royal jelly. A larva that had been fed on the worker-bee fare for a week or two already could hardly play catch up to join the royal ranks so late in the game. Queens must be fed more of this royal jelly from the early stages onward.
I have never observed worker bees taking any interest or initiative in destroying queen cells. Worker bees would be happy to use those cells, as needed, or leave them empty when not needed. I have often seen empty queen cells. The existence of the cell does not mean it must always be in use. If any member of the hive would have a wish to destroy them, it might be the queen, who does not desire competition. But queens do not do manual labor of any kind beyond their task of egg laying. The worker bees actually control or greatly influence even the egg laying--by how much they feed and pamper the queen.
[NOTE: I used to do beekeeping as a hobby, and before that I worked for a beekeeper--moving bees, splitting hives, requeening/adding new queens to hives, installing queen excluders to keep the honeycomb brood-free, extracting honey, etc. This knowledge of bees comes in large part from actual observation of them and from anecdotal evidences, rather than from scientific papers. If papers (second-hand science) are preferred here above actual observation (first-hand anecdotal evidence), then perhaps this answer may come up short of the site requirements. I am new here, and uncertain as to how rigid the rules might be; nevertheless, I am more familiar with the bees themselves than with papers written about them.]