Sorry I'm a little lost..I understand the one question of getting it, which, yes, if you already paid for your badge (pre reg) you may pick it up on day 0. However I don't understand the rest..It sounds like you are asking if you have to have your badge on you if you do anything on day 0. If this is a question you are asking, than no. Day 0 is not a convention day, the convention is not opened and operating at that time, so it's not needed. If they have ANYTHING opened (I heard on another thread once they had the vendors open that day, no idea if it's true, but if so, I'd assume you need your badge because it's still a "con activity". Another question I see in here is about cosplaying..You can cosplay all you want, not sure if that's your question, but no one can stop you from cosplaying that day, it's not against the rules or anything. And finally is a mix of both.."If I cosplay on day 0 do I need my badge" I see this being a possible question in your post too. The same as the answer to just having to have the badge. If you cosplay, you don't have to have your badge on day 0 other than if they, like above, open anything con specific.
Well I hope I helped. I probably answered things you didn't ask, but I didn't know 100% so cover all the bases haha