Thank you for the video and also for your service!! My son just joined the marines at 18 yrs old and also bought a 2007 c6 Z06 few months ago so now I can take care of it for him.
I recommend using the OEM AC-Delco ls 75W-90 gear oil which already has the friction modifier in the quart bottles, a Z06 will take a hair less than 3 quarts. Also, NEVER get under a car that is suspended by just hydraulic jack(s) use jack stands as a back up