But regarding a Sabbath act done by a small child, for example, I saw it written like this:
If a small child performed labor on the Sabbath, and he did it for the sake of an adult, it is forbidden to benefit from it on the Sabbath and after the Sabbath until enough time has passed in which that same labor could have been done.

But in this responsum you wrote differently regarding a Sabbath act done by a secular Jew:
https://mikyab.net/%D7%A9%D7%95%D7%AA/%d7%94%d7%a0%d7%90%d7%94-%d7%9e%d7%9e%d7%a6%d7%a8%d7%9b%d7%99%d7%9d-%d7%a9%d7%a0%d7%a7%d7%a0%d7%95-%d7%91%d7%a9%d7%91%d7%aa/

There you wrote that it is forbidden to benefit from his Sabbath act if it was done for an adult, even after enough time has passed in which that same labor could have been done.

What is the difference between a secular Jew and a small child, if neither of them is subject to transgression?

Reason suggests that the act was done as my agency and for me, and it is like an act that I myself did intentionally, which is forbidden forever (not necessarily by the formal law of agency itself). In the case of a child whom I sent to do something for me, one could say the same thing—unlike a child who does something for me without my having sent him. And perhaps with a child there is no agency, so there is more room to be lenient. But that is my own reasoning.

In connection with this responsum, I wanted to ask about telling a secular Jew to do labor on the Sabbath. Suppose the electric breaker tripped in the house and there is a secular Jew in the house who is willing to turn it back on and only asks where the electrical panel is—would it be permitted to direct him there? What I was thinking is that there is a difference between telling a secular Jew and telling a child or a non-Jew, because a secular Jew is a new reality that the Sages did not know, and therefore all their decrees about telling a non-Jew or a child do not apply to a secular Jew. In addition, in the case of a secular Jew the law of education does not apply, unlike a child. What do you think?

Turning it on for you? Then it is a Sabbath act. And from the standpoint of the decree against telling a non-Jew, it is all the more so for a secular Jew. This is an interpretation of the existing decree, not a new decree.

There is not really any direct instruction to turn it on here; I only told him where the electrical panel is. Is that kind of statement forbidden in the case of a non-Jew?

Not because of telling a non-Jew, but because of a Sabbath act.

Why is this a Sabbath act if the secular Jew is not subject to transgression (like a child), and the law of education also does not apply to him (unlike a child)?

It has nothing to do with education or with transgression. Even a Sabbath act of a non-Jew is forbidden.

But from what I saw online, in such a case it is permitted to hint to a non-Jew to turn the breaker on, no?

They permitted only a rabbinic prohibition upon a rabbinic prohibition in a case of a commandment.

Is there any distinction in a case where the secular Jew or the non-Jew or the child does the labor for his own sake as well (and of course also for you)?

If the non-Jew did it for his own needs, there is no prohibition of a Sabbath act. As for when he did it both for his own needs and for mine together, I do not have a clear answer. It seems to me that if he would have done it for his own needs even if I did not need it, then it should be permitted.

So if we go back to the original case where the electric breaker trips on the Sabbath and there is a secular guest in the house: the secular guest would be happy to turn the electricity back on for his own needs as well (air conditioner, hot-water urn, light, hot plate), and also for the rest of the household. If the secular guest is about to turn the breaker on, do you need to protest?

Here, by reasoning, it seems that one should protest. He is turning on my electricity in my house. He is only a guest. Again, in a rabbinic prohibition there is room to be lenient, since it is no worse than a rabbinic prohibition upon a rabbinic prohibition in a case of a commandment.
Along these lines, I explained that when Rabbi Shimon bar Nachmani seated Rav Huna and Rav Hisda in a sukkah that was invalid according to their view (Sukkah 10), it is a case of “two sides of the river,” because the sukkah is his and not theirs.

In general, do you have any advice what to do in a case where the electric breaker trips on the Sabbath? There is financial loss here, because the contents of the refrigerator and freezer will spoil without cooling. How would you act in such a case?

The prohibition against telling a non-Jew was permitted in a case of a commandment or great need. So there is certainly room to permit telling a non-Jew in such a situation. As for telling a Jew, that is more problematic, since there is room to say that when one actively feeds a prohibition, his act is considered your transgression (Rashi at the beginning of Mattot), and if he is acting unintentionally or under compulsion then even more so (Maimonides at the end of the laws of kilayim and elsewhere). As a general rule, there is a difference between causing a secular Jew (an atheist, or at least someone who does not believe in the revelation at Sinai and in being obligated by the commandments) to sin—where I leaned toward permitting it—and actively feeding him a prohibition and/or bringing him to commit a transgression for me. However, if he does it on his own initiative, as described above, there is no active feeding by hand here (as in the Mishnah, Sabbath: “If a non-Jew comes to extinguish, one does not tell him ‘extinguish’ nor ‘do not extinguish’”), because his Sabbath rest is not incumbent upon them, and he is not acting on the Jew’s instruction, unlike a child.

But above you said that they permitted only a rabbinic prohibition upon a rabbinic prohibition, and raising the breaker is Torah-level, no?

Regarding the prohibition of actively feeding a secular adult: does this prohibition of active feeding apply to adults at all? From what I saw in the sources, it was defined as a prohibition regarding minors.

A rabbinic prohibition they permitted. In a Torah-level prohibition one relies on the Itur in a case of great need.
The prohibition of active feeding exists with adults as well. Both by reasoning, and see Maimonides at the end of the laws of kilayim and elsewhere.

But if the prohibition of active feeding applies to adults, what is the difference between it and the prohibition of “do not place a stumbling block” or the prohibition of assisting?

Regarding kilayim in Maimonides, I assume you meant this:
If one puts kilayim on his fellow, then if the wearer acted intentionally, the wearer is lashed, and the one who dressed him violates “do not place a stumbling block before the blind.” But if the wearer did not know that the garment was kilayim, and the one who dressed him acted intentionally, the one who dressed him is lashed and the wearer is exempt.

But the prohibition of active feeding is not mentioned here; rather, specifically “do not place a stumbling block,” even in a case where the wearer does not know that the garment is kilayim.

“Do not place a stumbling block” is causing someone to stumble. Active feeding by hand is actively giving a prohibited thing. In the laws of kilayim, Maimonides is dealing precisely with that.

The opposite. Read it again. When the wearer is intentional, the one who dresses him violates “do not place a stumbling block,” and for that one is not lashed. When the wearer is unintentional, the one who dresses him is lashed for the prohibition of kilayim.

But for some reason the commentators on this law do not mention the prohibition of active feeding at all. See, for example, the Kesef Mishneh there:

“If one puts kilayim on his fellow, etc.” This is in the Tosefta in the chapter “These Are Those Who Are Lashed,” except that it appears to be written incorrectly and is not aligned with the words of our master. It is puzzling what our master wrote, that if the wearer did not know that the garment was kilayim and the one who dressed him acted intentionally, the one who dressed him is lashed. Why should he be lashed? Scripture does not say, ‘you shall not dress [another].’ And if because he violated the prohibition of ‘do not place a stumbling block before the blind,’ one is not lashed for that, because it is a general prohibition. And he himself wrote shortly thereafter, ‘he violates do not place a stumbling block before the blind,’ which implies that he is not lashed, for if not, what difference is there between the first clause and the latter clause regarding the one who dresses him? Another difficulty is that even if we were to say that ‘you shall not wear’ can be read as applying to him, as we say in the chapter ‘Three Kinds’ regarding a Nazirite, ‘he that shaves [another] is like one who shaves himself,’ because it is written, ‘a razor shall not pass over his head’—we read it as ‘he shall not let it pass,’ neither himself nor through another—still the first clause is difficult: why is the one who dresses him not lashed? This question is written in the responsa of the Rosh, who asked it of the Rashba, and we have not merited his answer. And I am surprised that the Raavad did not object to our master on this. However, this is not difficult: why is the wearer lashed even though he did no action, since another person dressed him? For one may say that it is impossible for another to dress him unless the wearer himself positions himself and adjusts his body, and that is considered an act, as we say in the chapter ‘These Are Those Who Are Lashed’ regarding one who has the corners of his hair cut. As for the earlier difficulty, one can say there that our master holds that since the Merciful One expressed wearer and dresser in one term, one is taken from the plain meaning and one is derived by interpretation, and both cannot be obligated together; whenever one is obligated, the other is exempt. And it is not comparable to ‘a razor shall not pass over his head,’ for there the derived prohibition comes not from interpretation but from the implication of ‘shall not pass’.”

From his words it appears that he did not raise the prohibition of active feeding as a possibility.

Exactly. That is why everyone gets tangled up in understanding his words (it already starts with the Rosh). But this is a distinction that Maimonides makes. And the Kesef Mishneh’s attempt to tie it to the specific law of dressing another in kilayim does not fit Maimonides, because he brings a similar law also in the laws of a Nazirite and priestly impurity. They simply did not understand that he is speaking here about the prohibition of active feeding.

Regarding active feeding with adults: in its original source, the prohibition of active feeding is learned in order to add a prohibition concerning someone who is not obligated in the commandments (a minor), when the prohibition is done by actively feeding the minor. I assume that this fits a mentally incompetent person as well (even if an adult). But from where do we extend this and say that the prohibition of active feeding also applies to someone who is already obligated in the commandments (an adult)? Is it a kind of a fortiori argument?

It does not add a prohibition to the minor, but to the adult. It is considered as though it were a prohibition that the adult himself performed. Therefore, according to Maimonides, the one who dresses another in kilayim is lashed, and not the wearer. The one who dressed him committed the prohibition. And for our purposes, Maimonides writes that if an Israelite renders a minor priest or an unintentional priest impure, he himself violates the prohibition of priestly impurity, even though the Israelite is not obligated in it.