You managed to get some of it but not quite all.
All three packs go off for engine start. Just allows more air from the APU to spin the starter motors, rather than trying to keep the cabin temperatures at a reasonable level
Above 300T, yes, it's a packs off take-off. Below 300T, packs on. All three packs get turned off for take-off. The packs are turned back on again once Climb Thrust is annunciated as the thrust mode. You put one pack on, let the cabin climb rate settle down and then put the other two on. Most people like to just pop on the second pack, wait a bit and then turn on the third but, strictly speaking, there is no need to wait between the second and third packs.
When is Climb Thrust selected? Depends on the airport and whether it's a noise abatement climb. If it is a noise abatement climb (as is the case in most places nowadays), then Climb Thrust is set at 1000'. If it's not a noise abatement climb then it's set when flap 5 is set. Obviously this is all set up in the FMC ahead of time, so VNAV is engaged after take-off (it's armed beforehand) and then Climb Thrust will automatically kick in depending on what's been set in the FMC.