If Rosh Chodesh is a "Chag" how can we start the additional level of mourning for the 9 days? On Rosh Chodesh during the Omer for example, one can (if I am correct) have a haircut, and the mourning aspects of the Omer are not kept. Why is Rosh Chodesh Av not like this?

The Shulchan Aruch (OC 493, 3) clearly states that it is a mistake to shave on Rosh Chodesh Iyar during the Omer mourning period (the Halachic authorities that permit it, don’t do so because of Rosh Chodesh rather because a different counting custom – see Mishnah Brurah and Biur Halacha ibid). Rosh Chodesh Av is the beginning of the sad month and as such we start the mourning practices from that date (ibid 551, 1), and just as a mourner doesn’t stop his mourning practices during Rosh Chodesh – we don’t too.