You start off by saying that normal would take some time before becoming equal to weight of placed body, this is wrong since the time for this would be so extremy small that you won't be able to measure with any possible clock.
1st : it is not accurate to say so, since normal force actually depends on weight of the body, although weight of a body is proportional to density, but while saying that normal depends on density do not forget it depends on volume also, so ultimately it depends on mass only.
2nd : as mentioned before the time gap in the magnitude of normal becoming equal to mg would be so extremely small, you should not bother about this time.
3rd : you are doing a basic mistake in applying both action and reaction on the same body, hence you are obtaining incorrect results.