#2 Split them and leave them as is for 4-5 days. Then you can go and look in the boxes and find the box with the new eggs in it and that is the one that has the queen.
How much time do you have, if you want to introduce a queen in the queenless box? Does it matter if they already started making queen cells? Will the new queen have the queen cells destroyed?
You can introduce a queen successfully up until a few days before the queen cells hatch. Once the queen shows movement within a cell the colony will return to queenright behavior. Generally that means that you can introduce a queen up until the capped queen cell is 7-8 days old. After that forget it.
No, an introduced queen will not destroy the queen cells, which means that if you have queen cells in a hive to which you intend to introduce a queen, it will swarm when the new queen hatches. If you have capped queen cells, or almost capped queen cells it might be a good idea to set up a nuc using those frames along with bees from another hive.
Newly hatched queens will usually go about locating and killing any unhatched queens but even then, more likely in multiple brood boxes, some cells may be missed which sets the stage for after swarms.
Hope that helps make things as clear as mud.