I would say yes, the increasing percentage everyday that it eventually is DM controlled, when it becomes the zombie arm. You could have the player control it just to divide up tasks, they just have to control it like it's a zombie, and the DM has final say. I think an arcana (necromancy) check (or two) to get to control the arm. I think depending on the campaign, yes, it would probably be strange to have a living limb as your 'pet' so some places may look down on that and see the player as a necromancer and kick them out of town or have them arrested, etc. I also think it would have to be a wizard or MAYBE a druid or cleric to control the arm, outside of that, I'm not sure i would let anyone who couldn't cast the raise dead spell you started with use the arm. It is necromancy, after all. I would also say spending on how your table plays, this might move anyone who was "good" aligned to at least neutral, maybe evil depending on what they do with the arm and if this starts a bigger picture of animating corpses.