In my experience both ducks and geese prefer to sleep in the open. One solution would be to build a secure, covered 'night pen' and lock them up each evening. They could be trained to enter the pen each evening with the use of feed. Even if they were to sleep on the island they would still be vulnerable to attack by great horned owls.
Izzy, people come first. Your instinct and reaction was to protect your mother and unborn sibling. That rooster was a deficit in your yard and needed to go. A quick blow led to a quick death. No one likes to kill, but when necessary it should be quick. If he were in my yard, I would have done the same thing.
Broody hens tend to become broody repeatedly - I have had hens raise as many as 5 clutches in a year.
In my experience flock population has no impact on the occurrence of broodiness.
Some people feel that the presence of a clutch of eggs encourages broodiness. I am not certain that this is factual.