My daughter is 10 months old, and my husband is fighting for shared custody. Given her developmental level and age, is this possible?

Of course it is. There is nothing that makes him less qualified than you to raise a 10 month old, fundamentally.

Yes, in fact because of her developmental age it is even more likely, since she will be getting the benefit of both parents.

Joint custody is very common and unless there is some reason against it, he will most likely get it.