A teacher colleague, who was also a mother of two students of mine, was murdered. Although she was not Jewish, can I say Kaddish for her and pray for her family?.
It is not recommended to say Kaddish in this case. You can pray for the family, and help them in any way you can.
A number of poskim, such as Zekan Aharon (2:87) Rav Ovadyah Yosef (see Yecheveh Daas 6:60; Yalkut Yosef 7:27), and others, write that it is permitted, in principle, to recite Kaddish over a non-Jew. This is also stated in Shut Maamakim (Vol. 3, p. 71). See also, at length, Mekavtziel, vol. 35, pp. 89ff.
However, the sources relate to special cases, such as a son born to a non-Jewish father, or to a non-Jew who saved Jewish lives in the Holocaust. In these cases, poskim write that it is permitted to say Kaddish for a non-Jew, and some write (see Zekan Aharon) that one should say Kaddish.
In the case of the question, you can certainly pray for the peace of the family, and for their wellbeing, and to help them in any way you can.
Concerning Kaddish, in the simple sense, Kaddish is an expression of honor for the departed, and applies specifically to parents. In a deeper, more kabbalistic sense, it is a tikun for the soul of the deceased, yet a tikun that is specific, in general to sons and descendents.
In special cases, a person can say Kaddish for non-family members, but extending this to non-Jews is not found in halachic authorities.
Therefore, it is not recommended to say Kaddish for a non-Jew, non-family member — although there is no formal prohibition in doing so.