At school when I drop and pick up my child am I allowed to say "Hi, how are you?" or introduce myself as the father of my child? I believe some scholars say even saying "Hi" is haraam. These are non Muslim women in a non Muslim country and they would feel this is cold or I don't want to mix with them. They may even feel as they are non Muslim that I want to stay away from them or think they are unclean. I think the possibility of fitna and so having zero interaction with them would do more harm than good. Their children maybe in the same class as mine for the next 13 years! Am I to not even speak to the mums (or acknowledge them) for this length of time? Should mums only speak with mums and dads only with dads?
I believe some scholars say even saying "Hi" is haraam
The scholars that are saying this are inciting fitnah and going against Islam. Allah says in the Quran:
Quran 4:86 And when you are greeted with a salutation, greet with one better than it, or return the same. Surely, Allah is the reckoner over everything.
Here, there is no condition on gender or religion. You can greet anyone you want.
Many hadith have been recorded in regards to the merits of offering greetings:
Abu Hurayrah said that the Messenger of Allah (PBUH) said: You cannot enter Paradise until you are a believer and your belief cannot be complete until you love each other. I tell you something which, if you put it in practice, will establish bonds of love among you all, and that is: Make salam a common practice among you which should include everyone, whether an acquaintance or a stranger. (Muslim)
Abdullah ibn Umar says that someone asked the Holy Prophet (PBUH): ‘Out of the practices of Islam, which is the worthiest?’ He said: Feed people and spread the practice of salam, whether you know or do not know a person. (Bukhari and Muslim)
Abdullah ibn Mas’ud reports that the Prophet (PBUH) said: Salam is one of the names of Allah Almighty with which He has blessed the people o the earth. So, make salam a common practice among you because, when a Muslim goes to a gathering of people and offers his salam to them, he is blessed with a station of distinction in the sight of Allah as he reminded everyone of Salam, that is, reminded everyone of Allah. If people in a gathering do not return his greeting, others will respond who are better then the people of this gather, that is, the angels of Allah.
Abu Hurayrah said the Holy Prophet (PBUH) is reported to have said: A big miser is the man who acts miserly in offering greetings. (Tabarani)
From all the hadiths and verse from the Quran mentioned above it is clear that you can offer greetings or answer greetings to anyone you want and there is no gender restriction mentioned in any of these hadiths or the verse.
Common sense and a clear understanding of the Quran should guide you in this case. There is no prohibition in the Quran for such interactions. And a quick glance at actual functioning Muslim societies reveals male shopkeepers or produce sellers interacting with housewives and exchanging money and goods. Extreme segregation can almost cripple a society and seems based on excessive mistrust between neighbors who are accustomed to social interaction that is beneficial, and certainly Muslim men are trustworthy enough. I believe the Quran gives us a certain amount of leeway in human relations rather than the current trend toward micromanagement and gender sequestration to the point of absurdity.
What matters is your faith and the quality of your relations. By interacting with others who are non-Muslim, you bring them your own attitude of Islam, based on the principles of faith, compassion, and having the ability to conduct normal interactions with people in a way that reflects your higher values. The norm in civilized societies is not rampant immorality but more cooperation and mutual trust toward a common goal. From Surat Al-Dharriat 51:23: 23. Then, by the Lord of heaven and earth, this is the very Truth, as much as the fact that ye can speak intelligently to each other.
The ayat that follow describe the "guests" of Prophet Ibrahim who were actually angels in the appearance of men, how Ibrahim didn't realize that but sensed that they wer "strange"— he immediately prepared for them a fatted calf, but they did not eat from it, saying they brought tidings of a son, upon hearing which Ibrahim's wife spoke up and laughed, saying she was old and barren.
Here we have our example, Prophet Ibrahim, sitting down to eat with "strange" men. His wife was there enough to have heard the conversation and acted as she would have with human men, perhaps not aware they were angels. There was no restriction for her presence at their visitation. Maybe he felt they were trustworthy, but what we are told is that they seemed "strange" to him to the point where he was afraid when they didn't eat. So by this example you can see that gender was not, in the home of that prophet, a matter of extreme segregation. Certainly parents of children in a school can converse without violating Allah's commands. It's the heart that must be pure, and capable of acting that way.