A woman’s haircut by a man and a man’s by a woman?

Is it permissible for a man to apply makeup to a woman for the above reason?

If that's his profession then yes.

Could the Rabbi please clarify the answer, ostensibly the entire art of the barber/makeup artist is to beautify the woman, so in reality he is preoccupied with her beauty, how is the permission of the one who is preoccupied with his art useful?

It has nothing to do with it. If this is his profession, then its beauty is a technical operation in his eyes.

1 So why are there rabbis who forbid it? They say that it is easy for a woman to go to a woman
2 How far is the boundary of a busy man in his profession? Is it permissible to go to a female masseuse if the client is a man (Swedish, Thai massage, etc. to relax muscles)

It's worth asking them. In any situation where the person's profession is at stake, this permit is relevant.

1 It says the site staff, who answers? Only Avraham Michael or are there other rabbis here who answer?
2 I would appreciate more explanation, please, how is it that in every profession the permit is relevant? It's just that a friend of mine studied treatments like Swedish massage and aromatherapy. Later, she converted. She also wanted to continue learning more massage methods and was also looking for a job, and almost every spa or hotel required that she work with men, so she couldn't find one. She also couldn't find non-intercourse studies (it's possible to find one, but it's far and difficult). In any case, all the rabbis she contacted said that it was immodest and that it violated the prohibition of touching because it's just a massage for general health and not some serious medical thing like physiotherapy. And she's fine with doing something that's not kosher if she can make an effort and find only female clients (but she didn't manage to open a business on her own). So does the rabbi believe that there's no modesty problem here? A problem of touching?
Doesn't the rabbi believe that a person should make a great effort to only work with men of their own sex? As if the rabbis actually told her, then give up if you can't do it, work with what you have, which is dedication to the word of God, and just as you wouldn't work on this on Shabbat, so don't work if you are a man. So what does the rabbi think about this?
Thank you very much

The titles are irrelevant. The answers here are only mine.

The permission is relevant in any profession when the concern is forbidden reflections. In Avidatiya Trid, it is said that there is no concern for such reflections. When it comes to an occupation that has an actual prohibition beyond forbidden reflections (and there are arguments about this in all sorts of contexts, such as touching and being alone. But in my opinion, in raids, both of these are also reflections), some have written that in this case the permission in Avidatiya Trid was not stated. Although I remember that there are poskim from which it appears that they permitted this as well.
Many poskim wrote this regarding gynecology and physiotherapy, and I see no difference between it and massage and the like. We are not talking about a permission due to necessity (which then would have been permissible where there was no acute medical need) but rather a permission from the law.
All of this is said according to halacha and the letter of the law. Whoever wants to be stricter before the Shweta will receive a blessing, but this is up to each person and their considerations.

Until we have to add here considerations of not being able to and not being able to intentionally do so in the first place, and so on.

To the Rabbi
Thank you very much for the answer (of course I would be happy if you could expand a little more or direct me to where to read what you said about the debates, whether it is also from a touching point of view (isn't it a problem that women are naked?) and specifically or just from a reflection point of view)
Thank you

I remember there is a book by Rabbi Haber on modesty, where he discusses all of this in detail.

Are massages that are completely nude also okay according to this explanation?

In principle, yes, as long as it is truly done for professional purposes. Although there is also the issue of modesty as a self-value, and according to this there is a problem here beyond forbidden contemplation and perhaps it will be forbidden for the patient (but not necessarily for the therapist).

Thanks for the answer
1 Regarding the book, I would appreciate it if anyone knows what the name of the book is, please? And what is the full name of Rabbi Haber?
2 I also wanted to ask about the patient's side, if it is possible that he has a prohibition, then is there no responsibility for the therapist on her side? Like before a blind person or some other prohibition?
Thank you very much and Happy Shabbat

https://tablet.otzar.org/book/book.php?book=163599
I don't think so, because if she doesn't do it, someone else will. At most, she'll help, and even that's highly doubtful.

Have a good week and thank you very much for the response. I will read the comments.

In the book Et Tznoim Hochma I did not find any reference to the issue, I would be happy if the Rabbi could direct me.
Thank you very much