Adding onto this question, right after Madoka recreated the entire universe to accommodate with the Law of Cycles, why was everybody placed back in time right after the event where Sayaka gets killed in battle? Why were they placed specifically there in time? Also, do all magical girls join the Law of Cycles after they die? If they do, then what is the point? Madoka's wish was to erase all witches with her own hands, so anybody else in the Law of Cycles wouldn't have a job to carry out since Madoka would be doing it all herself.
Right after Madoka recreated the entire universe to accommodate with the Law of Cycles, why was everybody placed back in time right after the event where Sayaka gets killed in battle? Why were they placed specifically there in time?
I don't think it's quite accurate to view everybody as having been placed back where Sayaka gets killed in battle. In this rewritten universe where the Law of Cycles exists, the entirety of history plays out differently, probably diverging whenever the very first magical girl's Soul Gem turns dark. Every scene involving witches or Madoka personally probably plays out differently; it just so happens that we only get to see one particular scene from this new history of the world - the scene where Sayaka dies. Why? Well, it's an important scene, particularly given how that same scene was something of a bloodbath in episode 10's timeline 3. And plus, it helps us understand how the new universe works.
I find it more likely that everybody was "placed" back at the same instant that Madoka made her wish in the main timeline. (I don't have any strong evidence for this; it just seems to be the most parsimonious option.)
Also, do all magical girls join the Law of Cycles after they die?
I'm guessing you haven't seen Rebellion.
Rebellion strongly suggests that the answer is yes, based on Sayaka's comments at the beginning of the final segment (shortly after Homura turns witch). Magical girls do indeed become part of the Law of Cycles when they're touched by Madoka when their Soul Gems go dark.
If they do, then what is the point? Madoka's wish was to erase all witches with her own hands, so anybody else in the Law of Cycles wouldn't have a job to carry out since Madoka would be doing it all herself.
Again, I'm guessing you haven't seen Rebellion, since in Rebellion, we learn that:
At minimum, Sayaka and Charlotte/Nagisa both sort of exist in the same metaphysical space as Madoka. Sayaka herself notes that the two of them are sort of like "secretaries" to Madoka. They both play a key role in Madoka's scheme to extract Homura from the Incubators' barrier; it seems unlikely that Madoka would have been able to effect her scheme without their (or at least somebody's) assistance.
First, note that at the end of the last episode, magical girls are said to fight wraiths (instead of witches), and Kyubey is still around (albeit with no confirmed knowledge of the alternative universe Homura is aware of). So the main things that have changed are:
What the magical girls fight. (Homura even states that humanity still faces certain "monsters", which just happen to not be witches.)
What happens after their soul gems become too "polluted". (In the new universe, Kyouko laments that Sayaka used up a little too much energy in their fight against some wraiths, and thus was absorbed into the Law of Cycles, just as they were getting closer.)
So magical girls are around, and it's simply that they don't become witches. There are thus no witches in this continuity.
After Madoka's actions, we see history rewritten entirely: in the beginning of the episode, we thus see her interact with the various magical girls who have come up through-out history, showing up to take them (which contrasts with the earlier scenario, where they were shown as girls who would later become witches). This also strongly suggests that yes, all magical girls join the Law of Cycles in this continuity.
As senshin notes, showing Homura in the "new" universe at the specific point where Sayaka dies serves as a good contrast with Sayaka's original fate. From Homura's possession of Madoka's ribbons at that point, it seems possible that Homura is placed back at that specific point in time in the new universe, but this is only in the sense that it's possible her consciousness was transferred to that point (unless, like with the numerous other universes Homura has experienced, she "arrived" when she was still in the hospital or at some other point, and it's just that she keeps the ribbons on her person). However, no one else knows who Madoka is, so we must conclude that Mami and Kyouko are merely part of the rewritten universe -- they aren't really "placed" into it.
Here's an answer to my own question that I thought up.
Since Madoka recreated the entire universe, it was probably her own decision to place them back in that moment of time. She wanted them all to still be together as friends and to help each other fight the wraiths, so she recreated the new world in that way to let their friendships still last and not have absolutely EVERYTHING be rewritten. It was placed right after Sayaka's death since Sayaka wanted to join Madoka in the Law of Cycles, so she joined Madoka after her death.