自由が丘の駅で、大井町線から降りると、ママはトットちゃんの手を引っ張って、改札口を出ようとした。
As far as I know, a dictionary verb form + と is used to mean "when/if", "whenever" (e.g. 秋になると、木が赤くなります (it's always like that)).
But the sentence above is about a particular day, a single occasion. As I understand it, they should say,
... 大井町線から降りた時...
or
... 大井町線から降りてから...
Why is と used in this situation?
「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:
