IIRC, they breed in BO slots but you have to have the character open/visible somewhere on your screen rather then closed and forgotten; which is how they'll breed for your captain.
This is correct.
They will breed in Bridge Officer device slots if your Status window is open to that BOff, so that you can see his device slots.
They will also breed in Bridge Officer slots if your BOffs are with you on an away team. This means that you can increase your breeding potential by finding a Ground mission and AFKing in it.
Remember, multiple tribbles can breed in the same space if you have only the tribbles that will breed and only the food that will cause them to breed what you want. This means that you can potentially stack 20 (cav)s and (dea)s in your bank with 20 Comet Cocktails and breed 20 Triolic Tribbles in just one hour.
It's worth repeating that they eat/breed on even-numbered hours played (type /played on a character to see how long you have played that character. On the even-numbered hours, tribbles with eat/breed. e.g. time played = 5 hr 42 min. You have 18 min until the next eat/breed opportunity).