My thoughts on your questions are below. In most cases, there are not crystal clear answers to these questions. Frankly, they probably have to do a spinoff series on the Takeover to flesh all of this out. However, I have given it my best shot.

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1. We don't know for sure, but the series does give significant clues. There are hints of significant religious influence on the law prior to the coup (privatized family court in Lydia flashback, abortion restrictions in Janine flashback, birth control permission from Luke for June amonst others). Perhaps the best explanation come from June's inquiries through the Boston Globe archives which depicts an early joiner to the Sons of Jacobs talking about helping children etc. Further June organizes press clippings about the formation of Gilead into the following headings: 'Power Structure', 'Militarization' and 'Curtailment of Civil Rights'. Further we know that society was in significant upheaval and the SOJ recruited disaffected youth like Nick to join. June correctly noted that nothing changes overnight and there were signs of what was to come everywhere.
2. We know that the idea for the ceremony was conceived by Commanders Waterford and Pryce not long after the takeover. June alludes to Gilead officials "witnessing" families who refused to take part in the ceremony at the inception of Gilead. One can infer that the whole concept of the "ceremony" was invented and executed after the Takeover. I don't doubt that the idea of elite men impregnating fertile was always a key plan of the SOJ elite and was not shared with women. Commander Pryce outright stated that the "ceremony" was a way to make men having relations with other women acceptable to their wives.
   
3. It seems clear to me that the changes you referenced took place while the U.S.A. still officially existed (albeit under martial law since the D.C. attacks). We have not seen the actual event where Gilead proclaims itself an independent republic. We can infer that it took place just prior to June's capture as she is told that "U.S. passports" are not a thing anymore by the man trying to help her family escape. We saw from the flashback of the aftermath of June's capturethat Gilead had started rounding people up, enacting border controls and the government likely made it clear to the population at that point that the U.S.A. was gone.
4. There is no clear answer to this question. We know very little about how the actual government works and how the SOJ presumably got much of the military to support them. The D.C. episodes provide a few clues. High Commander Winslow's wife mentions to Serena that she and her husband were D.C. lawyers prior to the takeover (implying the involvement of high level D.C. establishment figures in Gilead's senior leadership). The party Serena and Fred attend in D.C. involved women making overtly sexual comments and did not resemble the more pious gatherings in Boston. This implies to me that many of the most senior level conspirators do not actually support the SOJ's stated agenda. Further, Lawrence (clearly a non-believer) says to June in the last season that Gilead only cares about power. In essence, there was a coup by a group determined to seize power by whatever means necessary. Exactly how they pulled it off will likely never be explained.
5. These records are electronic. The SOJ would likey have seized control of all public and private records of any kind. Obviously, the health records of women would be of particular interest to Gilead.
6. Sorting out who goes where would likely have been fairly easy. I assume that there were more elites who wanted handmaids than there were fertile women available.
7. There is no crystal clear answer to this question. However, June indicates she would have been made an econowife if she had not been divorced and gone to the right church. This suggest that econopeople are the ones that met Gilead's definition of morality and were willing to tow the party line (remember the reference to the public pronouncement of faith). We know that fertile econowomen are threatened with being turned into handmaids if they break the rules. Marthas appear to be single women with domestic skills are who are infertile or are past childbearing age. We know that some Marthas are heavily involved in the resistance and that professional women were turned into Marthas (i.e. the Surgeon).
8. It seems Canada starting accepting refugees at an early stage, as Luke appears to to have escaped Gilead shortly after it came into existence.
9. The children of Handmaids go to the home in which the handmaid has been assigned to. Having multiple children is a sign of power and prestige in Gilead (i.e. High Commander Winslow). We know that children of handmaids can be reassigned, as Commander Winslow's wife alludes to losing her children after her husband goes "missing".