The stuffed animal costume on a V3 is the nyatchi (which I think is more of a cat than a teddy bear - but anyway its the character I have on my avatar). You can only get a nyatchi by wearing the costume...you can use it on any generation.
Even or odd generations just refers to the generation number you are on...if you look in your health meter (first icon) scroll through the screens until you get to the one that shows Gender and Generation. The generation number will be either odd or even 1,3,5 etc is odd vs 2,4,6 etc is even. And as TamagotchiLoverMB said - the link to the chart above shows what characters you can get with odd generations and which with even generations.
GK1