What are those buttons pushed after thrust is set to max
They're starting the chronometer. This chronometer is displayed on the navigation display.

It lets them keep track of the time limit they are allowed to operate at takeoff thrust. The limit is 5 minutes normally extended to 10 with an engine failure. After this, they must reduce power to the maximum continuous thrust setting (MCT).

There is also a time limit between the setting of thrust and when the engine reaches commanded power. If it is too long, it indicates a malfunction and takeoff should be aborted.