It says in Pirkei Avot to greet every person, and to receive every person with a cheerful face - but in the book of Vayikra, in Parashat Emor, when discussing the story of an israelite man who blasphemed the Name of HaShem - it mentions his mother Shlomit - Rashi comments that she was called this because she used to "chatter" Shalom Alechem to everyone. I read that her flaw was that she guilty of speaking with too many men and being too free with her greetings. It also mentioned that this is condemned in Ketuvot 72a. Is there a prohibition against greeting people too freely? In a country where you walk on the street and greet people who pass you by - is this the wrong type of behaviour?

Greeting people is a good attribute as you quoted from Pirkei Avot. Yet, the importance of modesty is even greater than that; therefore men should greet men and women should greet women, and doing so, fulfill both Pirkei Avot and Rashi's requirements. (For more see Kidushin 70b; Yabia Omer 4 YD 35; Rav kook, Mussar Avicha, Tzniut) More on the subject