I don't find the broodiness to be learned, but inherited. If your Silkie roo is carrying "broody genes" which is likely but not guaranteed some of the Leghorn Silkie crosses will go broody.
Is that what you are asking? Or are you wonder if silkie/leghorns will lay eggs? Different breeds of...
Chickens have a instinct to roost before it is dark. Do things happen that cause a bird not to go to roost? Sure any thing can happen when live beings are involved, but it would not be normal.
In those temps a chick that is starting to feather does not need heat. Some pictures of your setup are needed if you would like suggestions on how to integrate.
As long as the chicks have access to food and water and temperatures are reasonable (humans don't require coats) the chicks will survive on their own like brooder chicks. Some chickens survive without ever foraging. Your chick will likely venture out in a few days or weeks depending on your setup.
Of course the best companions for ducks are ducks, but sometimes that is not how it works out.
Between the geese and the chickens I would choose the chickens as long as there is space and you can get her water to dunk her head in daily. I have had some ducks get very attached to chickens even...
Beaks continue to grow. Before long you won't know it happened. If you can't get the broken part out of the way give him a brick to rub it off on his own
Any chance of putting the chick inside a smaller enclosure inside the coop or in the area the Bigs free range? If it were my chick I would work to get it integrated into the flock now. Don't expect the chick to be an active part of the main flock until laying age, but it might be able to live...
Be patient. Unless Mom has gotten up on her own, don't mess with the nest during hatch. Let her naturally hatch them. Interfering during hatch could affect humidity.
Be aware the hens might be aggressive toward her if you try to integrate her back to the flock later. Being able to see her does not hold her place in the hen order.
I frequently tell flock owners to feed all flock. It is nutritionally better than most layer feeds, but my birds still get layer feed quite often because there is quite a cost difference when feeding 70 birds. Personally I don't mix. If I have both I feed each one in different feeders.
The longer you get from hatch day the more likely Mom is to reject it. The injury might have happened when it didn't keep up or it might have been mom, but It is time for a new plan because now she does not consider it part of the brood. Taking another chick in to brood with it is a good plan...