What on earth is that sticky, stretchy stuff that Karura's eating? I've seen it in too many new year pictures now, and I want to know what it is.

Mochi. http://en.wikipedia.org/wiki/Mochi_%28food%29

There's a ceremony at new years where at the temples where everyone that wants to gets together and beat the shit out of a bunch of rice and they make mochi with it.

It's likely miso, given that that's what's mentioned in the text right next to her. Could also be soba, but I don't think you eat mochi with chopsticks...so that rules that out.

Mochi Is eaten with chopsticks as well. My family makes it every new year. Usually put it in a soup, or just eat it with shoyu and sugar.