You start off by saying that normal would take some time before becoming equal to weight of placed body, this is wrong because you are assuming that the bodies will sink in the ground due to their weight, but that is the practical case, in that case the tensile strength of ground gives in and it shows strain till it develops enough stress to counter 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 massof the body,and mass is a function of both density and volume of the bodh, remeber $ m = \rho V $ so as the volume need not be constant it is inaccurate to say that it depends on density
2nd : since in accordance with the realm of question, the bodies will not sink 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 have used $F_b$ applied by the person on ground in the equations for the block, this is not valid untill you are using 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 $F - μmg = ma$