Nope, the characters and their orders are the same every time.
Sorry if I'm asking so many questions but, does redoing orders you already had S's on affect the ranks in the gallery at all, because after going through Day 5 and the No Grav level again to get an S on Orva's and the Captain's orders respectively; I got Glorpina up from A to S rank from completing her order too which was already marked as an S rank while Orva and the Captain stayed at B in the gallery.
Edit: Just redid the first two levels which were already both at S's too and now Glubb and Orva are at A instead of B, Captain's still at B though
Edit 2: Redid Lights Out which also was already at S, and now everyone except the Captain is at S whereas he is still at B? Honestly I have no real idea what is going on with the averaging system anymore.
Redoing orders does not matter unless you score higher on them. The characters overall rank depends only on how well you do on their specific orders throughout all the days, the ranks for each day are averaging out the scores every character got on that specific day.
At this point the interaction between the gallery, the levels, and the weights on any particular save file depends on what scores you got on which levels in what order during which version and how many times you visited the gallery in between, which is an unfortunate consequence of me fixing these things over time while people are playing.
That being said, starting from a fresh save is the only thing I can bug test on since I don't have everyone's save files, but as long as you score majority S rank on a character's orders throughout all the days you should be able to see all their weight stages in the gallery.