Why do you think KATOU Ai is always with the Dokomo-dake mushroom family? She clearly isn't related by blood, and she never seems to be interacting with any particular one of them, so I don't even think she's that close a friend. The kids ignore her but she doesn't seem to resent them for it, so she's probably not adopted. (Big age difference, too.)

I imagine that she was one married to the eldest Dokomo-dake son, perhaps named "Chounan Dokomo-dake". Their love was fierce and tragic, opposed by the parents on both sides, but eventually the Dokomo-dake family gave in, realizing that their foolish prejudices had almost lost them their eldest son. Her parents, who came from more conservative families, never did see reason, and refused to see her even when her husband was killed in a freak accident at his construction job.

She only heard when she came back from her job making sandwiches at Kanda station. They were saving to buy their dream house.

Now the Dokomo-dakes are all she has left, and they're too good-hearted to throw her out... but deep down, they can't help blaming her for Chounan's death. They know it's unfair, but they can't let her in any more. And that's why she's outside making snowmushrooms while they're all warm in their igloos. Look how bravely she smiles.