「V(辞書形)＋と〜」 in that case is used to convey the meaning that the actions happen in succession, that B happens in the continuity of A, in the sense that there is no temporal break between the two actions.
In your case, "getting down from the train" and "taking totto's hand to go out of the station" happen in one go, there is no pause.
「V(て形)＋から〜」 「V(過去形)＋時〜」both carry the nuance 「Vが完了した後で、〜」, in other words, perform the second action after the first one was completed. The two actions are sequentially performed.
Hence, the alternatives you offer are in fact quite different nuance-wise. What is written in your book implies that the mom grabs her kid's hand "smoothly", instantly, as soon as the door opens.
In the two other cases, it sounds as if they:
- got down from the car
- (maybe there was a small break, maybe they looked for directions?) then, she grabbed her kid's hand