In a place where there is already light, it is permitted to tell a gentile to light candles for “additional light”…
Accordingly, since he won’t be creating any new action as a result of the air conditioner—he would stay in the room anyway, and even before this he could do so—it should also be permitted to tell a gentile to turn on the air conditioner (just as everyone agrees you can tell him to increase the air conditioning)… Just as we don’t look at each and every candle individually but at the situation before the candle was lit, so too one should not view turning on the air conditioner as a benefit that the Jew did not have beforehand, but only as an addition…
Isn’t that so?

First, that’s not correct. When the gentile adds light, it is permitted to use the added light, but there is no permission to tell him to light it even when there is already light, and likewise not to increase an air conditioner that is already running.
Second, intuitively I don’t think the cases are comparable. At the end of the day, there is no air conditioning in the room, even if I can remain there without air conditioning.

Of course I was mistaken—the intention was to say by way of a hint (describing the situation), and then it would be permitted to benefit….
If there is a fan, would the Rabbi agree that it would be permitted to tell him to turn on the air conditioner?

I didn’t understand. We already agreed that you were mistaken. You can’t tell him anything. The only possible question is whether it would be permitted to benefit from it if he did it on his own initiative. I don’t think a fan is just a small air conditioner, but it’s worth discussing.

I was mistaken when I wrote that it is permitted to tell him to turn it on…
But the halakhic decisors wrote that it is permitted to tell a gentile to do prohibited labor by way of a hint… not a direct hint, but “to say something from which he will understand that he should act,” to describe a situation and he will act on his own, because that is not considered that the Jew is commanding the non-Jew to do prohibited labor (Rema 307:22 and Magen Avraham, subsection 31, cited there in Mishnah Berurah)…

When it is forbidden to benefit from his actions, that won’t help, because even if he acted on his own it is still forbidden to benefit (for example, “my house is dark”); but when there is no positive benefit from his action, it is permitted (for example, “there is light in my house and it bothers me”)…
In our case, then, the question is whether it is permitted to benefit… If it is permitted to benefit, then it would also be permitted to hint to him (by describing the situation), and he would act on his own…

I don’t understand why to follow whether or not the object involved in the prohibition exists (the air conditioner, and whether a fan is a mini-air-conditioner)… The halakhic decisors explained the law that one may light an additional candle on the grounds that since he could have performed the same action beforehand, this is not “so much of a benefit” from the gentile’s action… (I think the logic is that in any case there is no need to decree against benefiting from a gentile’s action lest one come to command him—but my reasoning doesn’t depend on that)…

If so, one should always discuss the action he is about to perform: is it only made possible because of the gentile’s action? If it is possible to remain in the living room without the air conditioning, then this is not such a significant benefit and it is permitted to benefit from his action (or to “say something from which he will understand that he should act”)…
If it is extremely hot, such that it is very difficult to remain in that room, then indeed it would be forbidden (but then there are other leniencies, as you mentioned)…

The words of the Rema that you mentioned are puzzling, because elsewhere he himself writes that there is no difference between hinting and explicit speech, and he seemingly contradicts himself. Some have offered forced interpretations distinguishing between different kinds of hints, and I do not know the source for this (perhaps it depends on whether the prohibition of telling a non-Jew is because of “speaking of weekday matters” or because agency applies to a non-Jew stringently, but that can be rejected). And if there is a difference between kinds of hints, then it is not necessarily only when there is no benefit from his action (such as air conditioning after a fan), but rather in any situation it is permitted to hint indirectly.
As for your reasoning, I wrote that there is room to discuss it, and I am not settled on it. Even when there is no light at all one can do almost any action, but one cannot read, for example. So too with a fan: maybe one can live in the room, but not exercise there, for example. Even if your reasoning is correct, the line is a fine one.

If there is a difference between the hints, then it would be permitted to hint in any case by means of an indirect hint… but that still would not permit benefiting from an action that, had the non-Jew done it on his own, it would be forbidden to benefit from… Therefore it is a solution for someone who wants a gentile to turn off the light (it is permitted to benefit because there is no positive benefit, only the removal of something bothersome), but not to hint to a gentile to turn on the light, because it would still be forbidden to benefit from it… unless it is a case of additional light, where there is no prohibition of benefit, and the “problem” of saying it can be solved by hinting.