I'm doing practise questions for an upcoming test, and I have to name an interval from E to G double flat. Would that be a double diminished third? I'm not sure what to call it. Also if I invert it will it be a double augmented sixth?
You're exactly right on all fronts!
Since G♯ is in the E-major scale, E–G♯ is a major third. From there, we just make the necessary alterations until we reach E–G♭♭. If E–G♯ is a major third, then E–G♮ is a minor third, E–G♭ is a diminished third, and thus E–G♭♭ is a doubly diminished third.
In any event, it's definitely not a minor second (or even an augmented unison). These intervals are enharmonically equivalent to a doubly diminished third, but if the pitches are spelled as some type of E to some type of G, it must be measured as a third.
We can walk through the same process when we invert the interval, as well: G♯–E is a minor sixth, G♮–E a major sixth, G♭–E an augmented sixth, and thus G♭♭–E a doubly augmented sixth.
Intervals like this are really more "theoretical" in nature as opposed to "practical," but the logic in determining their size is sound (...excuse the pun).