Looks like running typings install github:typed-typings/npm-firebase && typings install (see issue would simplify the installation by completely eliminating Step 3 of installation. Also with this different firebase typings, I could also import * as firebase from 'firebase'; without any module not found error. The only downside is that the typings is referring to github until it's pushed to npm.
There's a issue tracking registry now. typings/registry#604