A mahram is an unmarriageable kin with whom marriage or sexual intercourse would be considered haram, illegal in Islam, or people from whom hijab is not obligatory.
This is a very important question, and relevant to all of us, as we all somehow socially interact with others. In today's society, the boundaries of non-mahram interaction are also weaker than before, which makes it even more challenging for us Muslims.
Please watch this presentation I gave in the holy month on this very topic, with detail explanation. It starts from the 1:38 hour mark:
If your adult daughters find it difficult to wear full Hijab at home in front of the Non Mahram man, you must protect them from falling in sin of being seem by him with out full Hijab. You must arrange for him another place to live, away from your adult daughters.
Mahram persons are those persons with whom marriage is always Haraam ( Not allowed for life) e.g. for male: His mother, his real sisters, his daughters, his wife's daughters from previous marriages, his real aunts, his father's wife, his daughter in law ( wife of his real son), his daughters' daughters, his sons' daughters.
Mahrams for female are e.g. Her father, her real uncles (Real brothers and real sisters of father or mother), her sons and their sons, her daughters' sons, her daughters' husbands, etc.
If the marriage between your father and her was consummated, she became your step mother and a Mahram for life even if she is divorced later on from your father.
If she has fulfilled her duty in Hijab, she will not then responsible but the man who looks at her with lust will be responsible of his evil thoughts. Men must avoid looking with lust to females especially who have done their full Hijab.
Muslim man is not allowed to meet and chat with women who are not Mahram for him unless in emergency or medical or security necessity. No friendship is permissible between male and female out of the discipline of marriage either permanent or temporary. If they go to public places with their families and they meet there by incident and with out planning, they should avoid chatting and mixing with non Mahram.
If they are mahram to each other then holding hands is permitted. So if they are 'engaged' meaning that they have either conducted a temporary or permanent aqd which has been read for them, then it's ok. If neither have been read for them yet, then they aren't allowed to physically touch each other regardless of what the parents say.
May Allah grant you success
In Shi'i fiqh, there is no problem with a woman travelling without a mahram.
In the Sunni schools of thought, travelling without a mahram is generally considered impermissible but sometimes allowed in some situations these days. So if she is looking for an answer in accordance with one of the Sunni schools of thought, she should consult someone knowledgeable about that specific school of thought.
If you definitely know that this female will use your beauty services in sinful way, then you should not help her in doing wrong. If you do not know that and they do wrong later on, you will not be responsible of their sinful acts.
In the case of other Muharrams, such as uncles , there is not much difference with others like father or brother, except for differences based on the culture of that community. But logically there is difference between husband and others.
Yes, a paternal and maternal uncle / aunt is considered mahram.
And Allah knows best.
Hugging in this kind of scenario would somehow entail emotional attraction, and a person of piety who observes the highest level of devotion to the Almighty does not want to engage in anything that would lead to haram or anything of that sort.
So, if both have decided they wish to get married, they should not delay the procedures for their Nikah ceremony, and after that they will be mahram to each other.
And Allah knows best.