During takeoff, why throttle forward then raise hand from throttle to push a button, instead of pushing button then throttling?
Actually he does push the button before selecting takeoff thrust.

Often1 the procedure on jets during take-off is to move the thrust levers about half way, wait for the engines to spool up and only then select TO/GA or Flex thrust as desired. In the video the pilot switches the display during this intermediate step. So the engines are not even spooled up yet and the aircraft is still moving slow. Plenty of time to stop if anything fails.

In fact even when nearing v₁ the reaction does not need to be that fast. The calculation of v₁ includes 2 s reaction time and correct response is more important. Look e.g. on this training video (it's A330, not A380, but the procedure is the same). The reaction is not that fast; they only retard the throttle after checking and calling the fault ("engine fire").

1It used to be that the engines could flame out if you added fuel too quickly, but that's no longer the case now when engines are managed by FADEC (full authority digital engine computer). Some manufacturers/airlines/pilots maintain the procedure, most likely to avoid applying full power if one of the engines manifests some problem early. Or at least the pilot in this video does it.