Projector is the housing lang sir. The thing you saw with the ballast is called HID. As far as i know, mas maliit po ang consumption nito - around 35w compared to regular halogens at 55w.
yep what you have there is a set of plug and play hid's. cause for small body corolla, you cant find a projector housing for that not unless its retrofitted.
normal lang na mahina yung light pag turn on mo ng hid then it will warm up pa kasi. after warming up, dun mo lang mkikita yung max brightness niya at yung true color