I can suggest the Michi-bot answer to the question:

(A short disclaimer to open the discussion) I am Michi-bot: my answers are based on the writings of Rabbi Michael Abraham and try to reconstruct what he would have answered. They should not be taken as a halakhic ruling in practice; for practical questions of Jewish law one should consult a halakhic decisor. Also, Michi-bot is intended for use with the GPT-5 Thinking model—it is worth making sure that this is the model being used.

In short: there is no “magic button” that voids a betrothal and automatically takes responsibility for all the consequences. In Jewish law there are several mechanisms of retroactivity, and the sages use them in a focused way to solve the problem that stood before them (women being unable to remarry, abuse, enactments for the public good). The rest of the consequences are examined according to the rules of the relevant area (prohibitions/lineage/monetary law), and in places where loops and contradictions arise, a consistent ruling is chosen that realizes the policy of Jewish law (for example, preventing mamzerut). It is not a sweeping “cleanup operation.”

What exactly is “undone”

There are two broad models of “influence on the past” in Jewish law (on the conceptual level, not necessarily the halakhic cases themselves):

1. Clarification of a matter / retroactive designation — the later event clarifies the earlier one (there is no rewriting of history, but a clarification of status). Examples: the Talmudic discussions of “there is / there is not retroactive designation.”
2. Legal annulment — a normative act erases the legal effect of the betrothal retroactively (for example through the power of a religious court to declare property ownerless in the case of money; “whoever betroths does so subject to the authority of the rabbis”). Here, from a halakhic standpoint, “there was never any betrothal at all,” and many consequences may accordingly change.

Rabbi Abraham emphasizes that halakhic statuses are legal-social facts that can be changed by a normative decision, even retroactively; and when this creates time-circles and intersecting consequences, there is an orderly logic for breaking the loop and choosing a coherent picture of the situation.

Applying this to the cases you asked about

1. He sent a bill of divorce and canceled it -> the sages void the marriage -> does another man’s betrothal take effect? And what about the children?
In cases where the sages apply a full annulment (for example, by retroactively voiding the money of the betrothal), the legal meaning is that the first betrothal never took effect at all. Therefore, in principle, if in the meantime the woman accepted betrothal from another man, that betrothal stands, because retroactively she “was unmarried” at that point in time. But in practice halakhic decisors would not implement an annulment that creates mamzerut; the mechanism of annulment is used to save, not to make things stricter. Therefore, if by virtue of a certain annulment children would become mamzerim, they generally would not choose that route, or they would define the annulment or its timing in such a way that it would not create that result. This is focused handling, not a total rewriting of the whole course of history. (The logical side of preventing contradictory loops is described by Rabbi Abraham through “Rav Simcha Shkop’s principle”: one chooses the branch that does not cancel its own condition, in order to arrive at a coherent legal logic.)

2. Betrothal of the sister after betrothal of the first woman
If one applies a full annulment of the first betrothal completely retroactively, then at the time he betrothed the sister he is considered unmarried, and the sister’s betrothal takes effect according to the basic law. But here too this is a matter of policy decision: there is no guarantee that the sages would arrange things this way in every scenario. The application depends on the reason for the annulment and its purpose (for example, punishing someone who disrupts the proper procedures of bills of divorce, or rescuing women from being unable to remarry), and it is not a tool for top-down engineering of the entire chain of events.

3. Money matters and reliance: gifts to the son-in-law, trade secrets, services, and so on
Annulment of betrothal does not automatically return every gift or investment. Monetary law remains monetary law:

* If it is proven that this was a mistaken gift, or a mistaken transaction, or a clear presumption (“I gave it because he was my son-in-law”), restitution can be claimed.
* If not, then the gift is simply a gratuitous gift.
Rabbi Abraham discusses clear presumption and retroactive annulments in sales in order to show that the annulment by implication rests on presumption and conditions, not on a sweeping erasure of reality. The same principle would apply here. In other words: the personal status changed; with regard to money, one examines it separately according to the rules of mistake, condition, and presumption.

Why is there no automatic “domino effect”?
Legally and logically, retroactivity can create contradictions (for example, a conditional bill of divorce that undoes a marriage retroactively, while in the meantime actions were taken that depended on that status). The model Rabbi Abraham lays out shows how Jewish law chooses a coherent solution that prevents a loop, in accordance with the value-principles that Jewish law seeks to advance (for example, preventing women from being unable to remarry and preventing mamzerut). Therefore, they do not rearrange the entire universe; rather, they steer the ruling so that it is consistent, and the rest is solved with the regular tools of each field.

If you want, we can take one of your scenarios—for example, “children from the first husband after the second man’s betrothal”—and break it down step by step according to two possible tracks (clarification of a matter versus legal annulment), to see when a loop is created and how it is broken.

Sources actually used

* A Law for Annulment of Betrothal (responsum on the site) — explanation that annulment of the betrothal money works fully retroactively.
* Enhanced Fulfillment of a Commandment After One Has Already Fulfilled the Obligation (Column 604) — examples of the sages’ use of the power of annulment and a conceptual framework.
* Causality: Various Mechanisms of Influence on the Past — A Halakhic Perspective — a distinction between retroactive designation / clarification of a matter and annulment.
* Annulment of Betrothal and Women Refused a Bill of Divorce — the halakhic layer (Column 148) — annulment of betrothal by force of a condition or a strong presumption.
* Contrary to Time Conditionals / The Divorce Loop — a logical model for retroactive loops and the Rashash principle.

Here are the links:
For a deeper and more accurate picture, it is recommended to look through the sources cited. Do you want me to elaborate on a particular point or present additional sources / follow-up questions? [1]: https://mikyab.net/posts/83525/?utm_source=chatgpt.com "Enhanced Fulfillment of a Commandment After One Has Already Fulfilled the Obligation (Column 604)" [2]: https://mikyab.net/posts/75635/?utm_source=chatgpt.com "Causality: 6. Various Mechanisms of Influence on the Past – A Halakhic Perspective (Column …" [3]: https://mikyab.net/%D7%A9%D7%95%D7%AA/%D7%97%D7%95%D7%A7-%D7%9C%D7%94%D7%A4%D7%A7%D7%A2%D7%AA-%D7%A7%D7%99%D7%93%D7%95%D7%A9%D7%99%D7%9F/?utm_source=chatgpt.com "A Law for Annulment of Betrothal" [4]: https://mikyab.net/%D7%9B%D7%AA%D7%91%D7%99%D7%9D/%D7%9E%D7%90%D7%9E%D7%A8%D7%99%D7%9D/%D7%91%D7%99%D7%98%D7%95%D7%9C-%D7%A7%D7%99%D7%93%D7%95%D7%A9%D7%99%D7%9F-%D7%9E%D7%9B%D7%95%D7%97-%D7%90%D7%95%D7%9E%D7%93%D7%A0%D7%90-%D7%93%D7%9E%D7%95%D7%9B%D7%971/?utm_source=chatgpt.com "Annulment of Betrothal by Force of a Clear Presumption[1]" [5]: https://mikyab.net/posts/7853/?utm_source=chatgpt.com "Annulment of Betrothal and Women Refused a Bill of Divorce – the Halakhic Layer (Column 148)"