Well the convention provide an ID badge which allows you into the convention and sponsor ones for the exclusive events. The character badges you see around are either made by artists themselves or paid for by those who don't draw. There is no hard fast rule on the size, but most people keep them no more then a few inches for space and cost.
You can check on Furbid or Furbuy for those offering commissions, FA or Deviantart are also both good places to find artists. Ask for reccommendations on those who are good. You can even just turn up at the convention and see who is offering for them to be made there and then or will do take home commissions that you can have posted to you later.