1) I am not aware of any mystical reasons against the haircut. The question involved is fear. However we see from the Shulchan Aruch Yoreh Deah 156:1 that one does not literally require a Reshus ha'Rabim to get a haircut; it is sufficient if it is given in a place which other people frequent. Most barber shops are in fairly public places.

In addition, the Mishnah Berurah 20:7 cites the Chayei Adam who writes that nowadays, idol-worshippers are not suspected of bloodshed. It is interesting that the Kaf Hachayim (whose author was born in Iraq in 1867) disagrees with the Chayei Adam and reports that there had been quite a few attempted murders even today, and therefore one should not have a haircut from a non-Jew in an isolated place. (Of course it all depends on the security situation in the particular locality.)

2) Regarding the Halachic definition you requested, possibly the simplest definition of an idol-worshipper is someone who does not believe in the Unity of G-d. Therefore Rabbi Moshe Feinstein rules (Igros Moshe Orach Chaim 1:50 DH b'Dvar) that even the group of Notzrim who believe the seventh day is holy and do not have idols or images in their places of worship, are nevertheless considered idol worshippers. Rav Ovadia Yosef (Yabia Omer II Yoreh Deah #11) also rules that one may not enter such places of worship.

However there is a Halachah (see Orach Chaim #156) that one may not form a partnership with an idol-worshipper because of the possibility that during the course of their business the partner may be obligated by the court to take an oath, and he will swear to his Avodah Zarah. The Jew thus will cause him to make an oath to Avodah Zarah. The Rema there writes that some are lenient on this matter nowadays because contemporary idol-worshippers do not swear by their idols. Even when they mention their deities, they mean the true Creator of Heaven and Earth. They merely combine the belief in the true G-d together with their own idol-worship.

KOL TUV

Dovid Bloom