ya i think Nike does get a roster for all 32 teams. Just like they have options for sizes for all 32 teams, they probably also have options for other things like jersey material (thickness of material, location of perforation and elastic locations). Players at different positions require different movement ability from their jerseys. For example, a QB will have different range of motion requirements than a kicker. I have seen different jersey types between different players who are on the same field. Marshawn Lynch usually wears a jersey that looks to be made of thin material, whereas russell wilson usually seems to wear a jersey that looks slightly thicker. Here is a picture of the NFL Nike Jersey Debut for the Vikings. The picture was taken by Nike and you can see that the jersey has cutoff sleeves that look pretty factory stock to me. You can also notice the alternate perforation design in this jersey.
I do not know specifically about the "elastic band" but I do think that the cutoff sleeve design comes from the Nike factory like that. In the LSU video I watched, the game manager was showing all the jerseys that were new from the factory and some of them had the sleeveless design. He also said most of the players prefer that design but didn't say if he actually made them himself or not. He noted that for players who took abuse (like the running back) they would usually get a new jersey after each game, whereas a punter would have the same jersey the entire season. The jerseys that LSU uses are the pro combat Nike Jerseys which are the same as the NFL jerseys. I assume it is the same procedure.
Now the case might be that the jerseys come from the factory with cutoff sleeves and the equipment managers add elastic bands if the players want but I still believe they come from the factory sleeveless. I was reading some forums online and they were talking about a few different sleeve options for players to choose from. That was for college Nike jerseys but im sure the same policy applies.
Quote from Authentic Game Memorabilia Forum:
"From the 2010 season, quite a few Cowboys skill position players moved to different cut of jersey, where the cuffed arm holes are part of the jersey body (like a lineman-style) instead of the cuffed sleeves worn previously. The same is true with the Nike jerseys, where the skill position players seem to wear two different cuts. It is the game issued jerseys with the cuffed sleeves that I wear during games, as the newer lineman-style game jerseys look ridiculous when worn without pads. (See this year's Superbowl media day interviews for proof!)"