A bypass capacitor on the input to the RPi might be enough, but it depends on the duration and magnitude of the dips caused by servos.
I would use an oscilloscope to investigate the dips. If they are very short (us) then I would add arbitrary caps (100uF, 1000uF) and re-test.
Another solution would be to try and confine those peak draw pulses by adding a diode from the battery to the Arduino, then connecting the Vin to the 5V regulator to before the diode. This way, transients won't pull from the input caps of Arduino, only from the battery.
It is possible that the question represents a misunderstanding or miscommunication of the kind of connection desired: Not something to be connected to the I2C bus itself, but to be controlled via some intermediary I2C to GPIO device, which further is connected on the I2C bus.
The idea of non-I2C devices being connected to an I2C bus is meaningless. The I2C protocol has specific requirements, for the I2C master to address the slave device, and for communicating data in either direction.
The LCD mentioned in the question, for instance, works on a different set of communication protocols, using multiple GPIO pins, and (from the description) most likely does not recognize I2C. There are however I2C LCD modules available, which incorporate I2C slave functionality on board - either through an LCD controller or a dedicated microcontroller with I2C slave communication code. Instead of the basic LCD module mentioned, such an I2C display module would be required for use on an I2C bus.
I2C bus expanders cannot magically create I2C functionality on a connected device. What I2C bus expanders can do, however, is provide additional GPIO pins, which are then controlled by code from the Raspberry Pi or microcontroller through the expander, which itself responds to I2C commands.
Perhaps that is what is desired - in which case, sure, it will work but not as described: A GPIO-driven device such as the LCD module or an LED can be wired up to the expander's GPIO pins, and the expander is the device that is connected to the I2C bus.
Best Answer
It looks like it should work fine at 3.3V. Each connection will draw about 3 mA of current from its RPi output. Yes, you need to connect the grounds together as well.
Gotta love the low-battery indicator circuit — it just wastes power until the battery is low enough to allow the LED to light up!