There are many ways to do this. I normally leave the hive queenless for a couple of hours, and listen for the “queenless roar”. At that point, I introduce the queen (in a cage, or using the Snelgrove matchbox method if I am in a hurry). If I don’t hear such a change in pitch and volume of buzzing, I wait until the next day.
Not usually. With africanized bees, you often do, and even then, they may kill the new queen. They would rather suicide the colony than accept a foreign queen!
That depends on a lot of things, but generally it is better to get her into the hive as soon as you can. The longer she goes without laying eggs, the more her ovarioles will shut down. She will start laying more slowly and may never lay as well as she would have if you had introduced her sooner.
To answer your question directly, I have kept queens alive for at least a week, if they have attendants. I usually give them a couple of drops of honey and a drop of water each day, keep them in a dark place inside the house (air-conditioned, but away from any drafts). The main problem is that the attendants start to die, and that can block the exit of the queen from the cage if you wait too long.