What breed of honeybee do you use?
We work with Buckfast bees. Check out our website if you would like to learn more about them: https://honeybee.uoguelph.ca/buckfast-breading/
How do you maintain genetic diversity/prevent inbreeding when breeding queens?
One way is buying new queens to increase the diversity of your stock.
How does creating a gentle hive affect their ability to defend the colony from invaders such as Varroa mites or robber bees? How does it affect honey productivity?
There is no evidence that aggression is linked to production. One dynamic that confuses this is that if the stock is somewhat aggressive a stronger colony will show more aggression than a weaker colony with the same genetics. The stronger colony may also be more likely to produce more honey and overwinter.
After replacing an aggressive queen, won’t her male offspring be left to mate with a virgin and pass down the aggressive genes again? What is the solution?
Unless you want to try to kill all the existing drones from the bad queen, you’re pretty much limited to just changing the queen out with a mated one so that all subsequent drones she produces are of her superior genetics. Drones are relatively short lived so the inferior drones from the bad queen should have a pretty minimal impact as long as the issue is caught and corrected ASAP.