In the name of Allah, the most compassionate, the most merciful
@user, you have asked many questions, but at least I strive to illustrate briefly (concerning the main issue).
From the view of Islam, there is no difference in the relationship of men and women (non-Mahram) between the real and virtual environment. Since the relationship in the virtual environment is like the solitude of two non-mahramsand. About the gathering which mentioned, logically if the gathering consists of solely 2 non-mahram, then it could makes the risk of sin, but if the gathering consists of more people, rationally the risk of sin reduces and probably won't be Haram. (of course by observing the Islamic points)
Of course in accordance with the rulings of Islam, it won't be haram if there is now probability of sin. Although we ought to pay attention that perhaps at majority of time (at first) there is not the intention or even the probability of sin, but later…
So, communication (or chatting) via SMS, email, facebook, or so on (with the mentioned condition which there is the risk of sin…), at many of times leads to the sin unless in some items. Seemingly, communication with non-Mahram, in the necessity or urgency situation could be permissible. Of course without the intention of sin or pleasure and also without using words which could lead to…
The view of some Maraje' of Shia regarding Chatting is that:
Whereas often the relation (internet or ways) between boys and girls
(non-mahram) will be the offspring of sin, therefore it is not
permissible and they should shun it.
For instance, look at the questions below which a person has inquired (which could help you):
What is the ruling of the chatting with the opposite sex by talking
concerning common matters?
Maraja' (of Shia):
It is not permissible if there is the risk or scare of sin.
Or for example you can see the response below which has responded to
another similar inquiry:
The view of Shiite Maraja':
… but in case of job relationship, no problem if it doesn't be the
cause (the introduction) of vice , of course by observing the Islamic
With respect to your last question which inquired regarding "will a Muslim woman be allowed to do Thawab(reward) by spreading Islam in a social gathering where both male and female are sat together", I ought to mention that logically it could be a positive practice and as a rule Islam will encourage these kinds of acts. Of course the significant point is that you should pay attention to the other side of the issue. Actually I mean you should notice if it could be the cause of "sin" in that environment or not (between males and females, by making a relationship between them…), If not, it will be appreciation from Islam.
But another important issue is that Satan tries to mislead from different ways, even though positive practices. How come? Let me give you an example. For instance, you as a male see a female who has no Hijab, So you really intend to talk to her to guide her …, but Satan will change the route of your goal … to be friend with her ...
On the whole, the important issue is that we ought to pay attention that Satan is trying to mislead us through various ways, then we take care about that.