I have read where breeders place a queen cell in a (hair roller type cage) at a certain stage of development and she hatches in this setup along with her "sisters." I understand this is done in order to protect her and the other queens from each other while all will be hatching about the same time. Is this correct?
Yes
If this is correct, how long will she, they, survive in this type of set up?
That depends on whether there are worker bees present. In an incubator, provided there is a supply of food in the roller cage, certainly a few days. But most breeders using this method check them daily at minimum. Some breeders add one or two worker bees inside the roller cage to ensure that the virgin queen is fed, although she's quite capable of feeding herself from honey placed in depressions made within the roller cage lid for that purpose.
Some breeders place roller cages around queen cells just prior to emergence within queenless colonies (not incubators) - the bees feed these virgins as they would any other bee in the colony.
Survival in a roller cage is not so much determined by feed itself - but by whether the virgin queen attempts to return into the queen cell. If she does this, then she'll not be able to back-out of the cell afterwards, and so will die inside it. Hence frequent inspections by the queen-rearer, who will 'squish' the wax cell to prevent this from happening.
Another question along these same lines. Once she is hatched, how long is the maximum amount of time that she can be kept from mating without ill effects on her reproductive abilities?
Emerged, not hatched. The figure usually given is around a month (if memory serves). After that time she is destined to become a drone-layer. Of perhaps more importance (imo) is what happens within the first few days: when the virgin first emerges, she produces no queen pheromone - and so can be placed with relative impunity into any colony without encountering issues of acceptance. However, after 3 days she will begin to smell like a queen, and so suitable introduction precautions then need to be taken.
Now the mated queen question. What is the longest that a mated queen be caged without ill effects on her laying ability. What is the time period in which she is caged and not afforded the opportunity to lay, without damaging her reproductive abilities?
This question is perhaps best addressed by examining experiments which have been conducted into the over-wintering of mated queens. It has been shown that as long as six months can be achieved (similar to the length of time that many colony-based queens spend without laying) - but - the issue then becomes one of survival itself. The figures for this are very poor (imo); with only 10-20 percent or so of banked caged queens (i.e. not colony-based) surviving for that length of time. In the studies I've read, subsequent laying performance appears to be unaffected - but it's the mortality figures themselves which I find so disappointing. In contrast mated Queens held in 5-frame nucs, and which encountered a similar period of non-laying, faired a lot better - 70 percent plus, if memory serves.
I could dig out the relevant papers if you want to delve deeper into this ...
LJ