No. That is not an injury that geese are capable of.
However, many predators can and do kill that way, then eat the head.
Any canine, large owls, weasels, etc.
Great news about Lucky.
You are correct, if she is no longer sitting the other eggs will not hatch.
I have an incubator in the house, so I put unhatched eggs in there when mom leaves.
You will need to give him a good once over... see if the mobility issue is a deformity... or ???
With late hatches... left behind eggs... . I usually keep them in the house until the "left over" eggs hatch AND the new hatchlings are running well. Then I put the new ones in with the others and...