These are vey good questions and I do not have all the answers, but one thing I wish to clarify as this misconception is quite widespread: this custom was not invented by Hassidim. In fact, this custom predates chassidus and possibly dates back to the days of the rishonim. See for example Shut tashbatz part 3, 299, where he mentions a custom of קציצת הפיאות כמשפט הבנות on the day the girls get married. However, this could simply be interpreted as a reference to a practical haircut that newlyweds used to get, and not a complete shave. But if it is indeed a reference to the shaving custom, then it is probably the earliest source to mention this (unless we count the Zohar which is unlikely to have been followed at that point).
(The passage in the Rambam cited by Robert S. Barnes forbidding Jewish women to shave can further be interpreted as a polemic against his fellow Sephardic Jews who followed this custom. In fact the Rambam himself demurs to a similar custom in Egypt, where the bride would dress like a male and dance with a sword under the chuppa in front of the men. This letter has been cited in full in the beginning of the sefer Maaseh Rokeach. The shaving custom if followed by Sephardim in the days of the Rambam may well be linked to this cross dressing theme/ritual observed by Egyptian Jews as the Rambam's letter attests to.)
Fast forward a few hundred years we find the Chasam sofer mentioning this same custom in his responsa YD (siman 195) in which he discusses the problems that arise from it in relation to tevila/immersion. The Netziv mentions this custom in his sefer Haamek Shaala part 12, 95 (responsa) as well, and calls it a custom that was "instituted by the Rabbis" suggesting that this custom is quite old. There he also discusses why this custom was not followed in the times of the Mishna. This custom is also briefly mentioned in the sefer keli chemda.
By now it should be clear that this custom is not exclusive to chassidim (chasam sofer and netziv were not chassidim I assure you), and that it also predates the establishment of chassidus. For some reason (most likely because it smacks of superstition) this custom was completely forsaken by the majority of Jewish kehilot, and was only preserved by a handful of Chassidim and by some yekki communities, but as I have shown, this picture does not do justice to the reality in the old European communities where this custom was widespread.
According to R Menashe Klein (in mishneh halachos my main source here) this custom was instituted by the Council of Four Lands. However this does not necessarily preclude an earlier date for this custom, as they may have merely been the ones who standardized it, but the custom may have been around well before it became widespread in Europe. You can read here for more on this, and the reasons behind this bizarre custom.
The most likely origin for this custom is the Zohar. In two places (Parshas Naso and Pinchas) the Zohar encourages women (particularly menstruating) to shave. In one place the niddah is encouraged to burn her hair because the hair is contaminated and dangerous. In another place the niddah is encouraged to shave before she immerses in the mikveh, so she becomes more pure and fit for her husband when she immerses. So the origin is most probably kabbalistic.
Before I close this discussion I would like to note that this custom is also found by the ancient people of Sparta; they had a similar custom of shaving off the head of the bride that was about to marry, and like the Egyptian Jewish brides, mentioned earlier, they dressed like men on the day of the wedding as well, see here.