"He opens the box."
Translation:Hij opent de doos.
I just asked a Dutch person this and it seems that they generally wouldn't use 'opendoen' for all objects, but mainly doors, windows, gates, etc.
Hij doet het raam open. Hij doet de deur open.
Another similar verb is 'openmaken', which you would use for other objects, such as letters, bottles, boxes, etc.
Hij maakt de fles open. Hij maakt het briefje open. Hij maakt de doos open.
It seems to me that part of the difference is the permanence of the 'openness'. A door and a window would be shut again ('opendoen' can be also be used for similar things such as 'monden' and 'boeken'). A bottle or a box might be opened to get the contents out and then discarded afterwards.
My dictionary (van Dale) also seems to suggest that 'opendoen' implies that something needs to be 'unlocked' ('van slot halen ook') to be opened, which could apply to a 'deur' and a 'raam', but not a 'doos'.
My Dutch friend also suggested that 'opendoen' is used for things which are opened more habitually/frequently (doors, books), but I guess this concept is very closely linked to the first idea of whether an object is likely to be closed again or not. A single box is not something most people open or close very often, but a door is.
'Openen', on the other hand, seems to be universally applicable.
In any case, it is probably something that needs to be learned by exposure to the language and if unsure you can always just use 'openen'.