LaNoZ

You could be right then again the show doesn't really care about the "real laws ". Emily/Amanda's birth certificate has footprints on it so it's possible that Margaux wanted to publish it to confirm her story.

I think I have an answer for the importance of the birth certificate for Margaux, since the original document had baby Amanda's footprint on it, she can prove that Emily is in fact Amanda. I'm maybe wrong but I recall reading somewhere that the whole purpose of footprints on the birth certificate was to be sure that the baby has a "legal" trace, in case of kidnapping for example (it was before dna, of course.)

I have a theory about Margaux's accident. Perhaps after the ultrasound, she realized that she wasn't caring Daniel's baby so she conveniently used her altercation with Emily hoping to lost dramatically the baby to have Victoria on the side. In the beginning of season 4, Daniel and Margaux weren't a couple (she wasn't upset when she found out that his one-night died in his hotel bed ) then after they framed her brother, they're kind of official until he cheated on her so it's possible that she slept with another guy by revenge. When she was aware of her pregnancy, she assumed it was Daniel's child.