I'm pretty sure she does carry it around, she's not like Nanao . Yachiru's Zanpaktou is adorable the guard looks like a flower and the bottom has little wheels on, because it drags on the floor when she walks .
As for if she knows it's name or not... That is a bit harder to say, as we've never seen her so much as draw the damn thing... But I think(hope...) we're going to real soon!