do you use the same aircraft for a 20-30 minute hop?
yes
does it still have a rocketship departure in the 40degree heat?
It probably weighs around 40t less even with round trip fuel. Getting airborne at 2000Lish and subsequent flying during cooler temps.
Similar shuttles are AUH-MCT and ANU-UVF/TAB/GND. ORD-IAH is longer at around 2 hrs
does the same crew do Heathrow - Cairo - Heathrow in one day on the 777?
Nope, arrive late Day 1 depart early Day 3. Aircraft is on the ground longer.