How should the behaviour of men be towards women; for example, can men speak with women, or can men shake hands with women, or can men salute women? Can men do these things ‎in any situations and what are these?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

The behavior of men towards women should be limited by Sharee’ah criteria and none should transgress the set limits of Almighty Allah or violate His sacred limitations.

As regards men shaking hands with non-Mahram women, it is forbidden. The evidence for this is sound and clear. ‘Aa’ishah said while telling the story of the women who gave pledge to the Prophet : “No! I swear by Allah, the hands of the Prophet has never touched the hand of a woman at all.” [Al-Bukhari and Muslim]

The Prophet Muhammad would shake hands with men who gave their pledge to him. Allah Says (what means): {Indeed, those who pledge allegiance to you, [O Muhammad] — they are actually pledging allegiance to Allah. The hand of Allah is over their hands.} [Quran 48:10]

There is no doubt that some women gave pledge to the Prophet but since he had not shaken hands with any of them as he did with men although there is the above verse - this means that it is not lawful to do so.
Another evidence is that the Prophet said: “If one of you were to be stabbed in the head with an iron needle, it would be better for him than touching a woman whom he is not permitted to touch.” [At-Tabaraani]

Moreover, it is not logical that Allah would forbid looking at women and allow touching them and shaking hands with them. As for talking to them or greeting them without shaking hands with them, then if this is done in a legal way and for some sound reasons and far from any doubts, bad words, or obscenity, then it is lawful, otherwise it is forbidden. There is a sound Hadeeth narrated by Ahmad in his Musnad and Ad-Daarimi in his Sunan that Prophet Muhammad once went past some women and greeted them, and they replied to the greeting. There are many other sound Ahadeeths proving that the Prophet addressed women and they addressed him too. There is also sound evidence that his Companions did so as well.

Allah knows best.