Is there any rule or custom against celebrating a bar mitzvah on rosh hodesh Av, considering that the beginning of Av is considered a time of mourning?
Nitei Gavriel Hilchos Bain HaMitzarim Volume 1 40:14 brings in the name of the Yad Efraim and Magen Avaraham that one may have meat and wine at a Bar Mitzva Seuda during the nine days. He goes on to say (Magen Avraham, Chayai Adam, Yad Yitzchak) that if the Bar Mitzva boy gives a Drasha it is permitted to have meat and wine even if it is not the actual day of his Bar Mitzva. However there are those that disagree (Yad Efraim) and say if it is not the actual day of the Bar Mitzva one may not have wine and meat.
Regarding Erev Tisha B'Av he brings from the Divrei Yatziv 50:3 that those that celebrate the morning when the boy puts on Tefilin, it is best to have the Seuda after Tisha B'Av.
I have personally attended Bar Mitzvas in the first few days of the month of Av. There was no music at these affairs.