I hold down the LEFT mouse button to move the camera (I'm tank so I need to keep an eye on the battlefield at all times). When I need to move quickly I just hold down the right mouse button in addition to the left button, and I'll be moving in the direction my camera is looking at. Very convenient, and I don't believe I experience the problem described in the op.

As for Q and E...I rebind A to Shift-A, S to Shift-S, then Q to A and E to S. So my default left-right movement is strafing already.