If it's the official 7" touch display, it's not HDMI, it uses a dedicated ribbon cable connector for everything except power and I2C to the touch controller. The I2C signals are on pins 3 and 5 on the header.
The MB6HC does not use I2C (it uses the power pins and pins 17, 19, 21, 22, 23, 24) , so it's safe to use the sort of expansion header you identify, yes.
You could make up your own connector that runs just the required wires to the MB6HC and required wires to the display. This might be tidier. You only have two 5V wires to play with, however. Also, I'm not sure about available power - If doing this I'd be inclined to power the Pi+display separately from the MB6HC. You'll still need to connect some 0V, but there are plenty of them on the Pi header - just connect any of them to the MB6HC and any other one to the display.
