The Canon EF 100-400mm f/4.5-5.6L IS II USM lens is the "Mark 2" version, and have 16 years worth of improvements on the older EF 100-400mm f/4.5-5.6L IS USM lens.
Both lenses would indeed suit your camera.
You can use these lenses with the 2X extender, however you may find that performance is limited due to diffraction. It should still be good but don't expect wonders.
The battery problem is common on these Canon cameras.
To resolve the battery problem, try this:
Remove the lens and put a cap on the body.
Turn off the camera but leave a battery in it.
Put the camera in a freezer for 6 minutes.
Then remove the camera and put a room temperature battery in the camera.
Turn on the camera.
The camera should see the battery and display the battery life.