Goran is absolutely correct with most of what he posts. Some will argue that 'nice' queens can be reared by tiny colonies, but that is not worth the risk, particularly for multiple queen production.
Re the feeding/finishing of queen cells in a queenright colony, think here of queen pheromone, demaree or queen loss. An old queen in a large colony may be superceded due to sparse queen pheromone; moving brood further (and further) away from the queen leads to queen cells being developed; no queen pheromone (lost queen) immediately triggers emergency cells to be developed.
The first emergency queen cells sealed are often built over older larvae, so are culled leaving those developed from the youngest larvae.
That is why multiple queen cells can be developed very easily in large crowded boxes which are hopelessly Q-.
After sealing, or at the point of sealing, transferring to smaller units is a better use of a queen rearing set up. For only one single batch of cells, it is clearly not as important. Think here - those rearing large numbers of queens on a serial basis may well place sealed cells into an incubator for pupation, so no bees being used at that stage.
As you are aware, the first queen out may kill all the rest. Therefore to rear multiple queens they must be kept separatedvat emergence. Those with incubators take suitable precautions (they will have many queens emerging almost at the same time) and others will place each sealed cell into a small box with sufficient bees to keep the pupating queen warm. That small box could be with just a cupful of bees up to a nucleus hive. The smallest boxes (apidae) are used until the queen is mated, after which she would either be used elsewhere or sold.
Queens will eventually be placed into a hive with sufficient bees to allow rapid colony expansion. If selling commercially, the beekeeper wants the most efficient use of the available resources (the best workers:queen ratio) whereas an 'end user' queen rearer must have sufficient bees available, to allow those reared queens to expand into usable units.
All queen rearing works on the same principles, just different methods are used, dependent on the intensity of production. You should pick a system which suits your resources and needs.
Even when I had 25 colonies, or more, I was able to easily produce sufficient queens (for my own use) without resorting to more than demaree-ing and splitting colonies into nucs, each with a good queen cell.
With only three colonies, your problems might be choosing a good queen from which to rear your next generation and having sufficient bees to make up nucs without weakening your production colonies
(choice of more colonies and less honey, perhaps?).
Not all queens are good ones. Some can lead to 'poor temperament' colonies, as an eample. Those
queens need to be culled and the bees either requeened or re-united with other colonies. Such is beekeeping that arriving at queens with all the required traits means careful selection of breeding queens and destruction of some of the reared queens - unless you have dedicated breeding lines set aside for just that purpose.
RAB