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.
Don't invite her alone to your house but try to do Da'wah in the presence of other persons.
Woman can go out with out Mahram as far as she is safe. Permission of her husband is required when her going out might contradict her duties as a wife.
Yes it is permissible for her to recite Tajweed or Tarteel of Quran in front of non-Mahram men as far as there is no intention of lust.
Female doctor or medical student is not allowed to touch a male patient unless in emergency.
Yes, there would not be any problem, as long as it is in the area a woman is allowed to show, and not a part of her hijab, and there is no other haram act involved, like lust, etc.
And Allah ta'ala knows best.
Hijab rules must be fully observed while meeting or talking to non-Mahram relatives. Islamic rules for non-Mahram are the same whether relative or not despite culture in some societies which do not follow Islam. Muslim should keep Silat Al-Rahim with all his/ her relatives with observing Islamic rules of Hijab.
Yes (if the narrations are correct).
Your husband's step father is non Mahram to you and to your future daughters as well, simply because he is not the biological father of your husband.
Both looking at non Mahram woman with make-up and mingling with her, are not allowed for Muslim man.
Muslim man should lower his gaze and avoid looking at non Mahram women especially when such looking moves his desire in any way.
It is not permissible because it is a type of touching with cloth with a non Mahram.
Daughter of your half brother is a real niece for you and she is a Mahram for you like your daughter. Your half brother's mother is your father's wife, who is also a Mahram for you like your real mother.
Step-dad of your husband in non Mahram to you and to your daughters. If he has a real son who can enter with you in a verbal temporary marriage contract for a short while with out consummating it , that will make his father (the step-dad) Mahram to you and your daughters. If that verbal contract is done with your daughter, his real father ( the step-father) will be Mahram to your daughter.