If I draw a rectangle, then extrude it, will it be identical to a box? I mean, will bottom face be closed as in box?
It will not be identical to the Box primitive, as the result will be a Rectangle with an Extrude modifier.
It may have the same apparent geometry like a box, assuming that you toggle the flag to cap the beginning and end of the extrude.
The origin of the resulting object will be at the base, which may differ from where the origin of a box normally is.