Regarding the weight variable used for analyzing the child weight, it is depending on the background. If you take only the children living with an interviewed mother, it is better to use the variable V005, the child's mother's weight. If you take all children measured in the household, it is better to use the household weight HV005. Regarding the repeated values of the weight variable (either HV005 or V005), note that the weight is a cluster level weight in DHS surveys, all HH in the same cluster share the same weight; all women in the same cluster share the same weight. So this is not a concern.
Thanks
Ruilin

For Q#1: The answer is no. The "re-normalize" (rather d-normalize in our words) is to reverse the normalization procedure, so it has to use the women population size and sample size. The children's population size and sample size were not used in the weight normalization.

For Q#2: The answer is no. As stated in the question above, "re-normalization" is to remove the normalization factor from the normalized weight, there is no other way. But if you pool different survey data from the same country, you can "re-normalize" the weight using number of households to approximate, this assumes that the average number of women 15-49 per household is constant across surveys.

V005*=V005×(total number of residential households in the country at the time of the survey)/(number of households interviewed in the survey)

Suppose some variable you are interested in had only been collected in, say, a survey with 1000 cases in Peru in 1990 and a survey with 5000 cases in Kenya in 2010. Would you combine them into a single file with 6000 cases and calculate the mean of the variable? Would that mean be interpretable?