I didn't see where he explained why the PNF (apparently mistakenly) thought the aircraft had achieved the positive rate of climb necessary before retracting the landing gear,
Numerous possibilities.
Getting to be too many pages to read back thru so this may have been mentioned; just a small thought, but why did ATC feel an urgency to transmit GA instructions to an aircraft that's in a dynamic maneuver at a critical phase approx 100' agl, and why did the PNF feel an urgency to reply?