I don't think bombs can be skipped, dodongo needs to be killed.
And I must agree a run skipping the sword will be more entertaining/surprising.
As for the slingshot, it's skipped.
Edit: Bombs cans be skipped it seems
http://www.youtube.com/watch?v=T_bO2F7fpSM
Edit 2: Sword can not be skipped since Mido blocks me from going to the Deku tree, and I need the spiritual stone to trigger the OoT-in-the-moat cutscene. If anyone has any idea as to how the sword skip could be achieved, feel free. In the meantime I'll continue the test run with the sword and without deku sticks.
Edit 3: I'll post something when I'll be out of the forest (fairy ocarina will be skipped as well)(thus magic meter will be taken as adult). Right now I beat Gohma slingshot-less-style and I'm in Lost woods.
G2g to work, be back in the morning. (3:30am east. time)