You start off by saying that normal would take some time before becoming equal to weight of placed body, this is wrong sincebecause you are assuming that the time for this would be so extremy smallbodies will sink in the ground due to their weight, but that you won't be ableis the practical case, in that case the tensile strength of ground gives in and it shows strain till it develops enough stress to measure with any possible clockcounter balance the total weight, the question you are asking is idealisitic and in this case the bodies will not sink in the ground and will remain above the surface of ground at all times.
1st : it is not accurate to say so, since normal force actually depends on weight ofmassof the body, although weight of a bodyand mass is proportional to density, but while saying that normal depends ona function of both density do not forget it depends onand volume alsoof the bodh, remeber $ m = \rho V $ so ultimatelyas the volume need not be constant it is inaccurate to say that it depends on mass only.density
2nd : as mentioned before the time gap since in accordance with the magnituderealm of normal becoming equal to mg would be so extremely smallquestion, you shouldthe bodies will not bother about thissink into the ground, it can be said as soon as they touch the ground the normal equals their toa weight without any time delay.
3rd : I see that you are doing a basic mistake in applying both action and reactionhave used $F_b$ applied by the person on ground in the same bodyequations for the block, hencethis is not valid untill you are obtaining incorrect resultsusing for the entire system at once, which wouls also include the reaction from ground to $ F_b$ applied by the person. Individually if you see the reaction from ground would equate $F_b$ the force $F$ applied by person on block and block's reaction $R$, then the rest can be equated as $F - μmg = ma$