I believe this scene is seen from Elias' perspective. There's a scene directly after Elias runs out of the room just showing the vacuum cleaner on the floor switched on with nobody operating it, which I believe is when it switches back to the 'real' perspective.

How could "Lukas" be vacuuming? Do you mean we're seeing this scene as imagined by Elias?
We know her only as "Mutter," per both the movie and the cast list here. What else to call her?
This scene puzzled me, too; I expected her to yell at E to go back to his chores.

It was Lukas holding the vacuum cleaner (he is always the one wearing darker colours, up until the scene where they decide to dress identically; he also has a mole on his neck that he can be easily identified by).
By the way, I wouldn't use "Mutter" like it's a name. It just means mother, it's not a character name.

He was clearly looking for the cat. He kept saying "Leo" too.

I sorta wondered about this scene also. It had to be Elias in front of her as she walked in the room, so that would mean the vacuum cleaner was lying on the floor. So either it wasn't lying there at all, or Mutter simply didn't see it lying there when she walked in and was focusing on Elias.