Say I'm given the notes like this.
How do I determine whether it's already in root, 1st, 3rd, or 7th inversion?
I'm going to disagree with Tim (he removed his answer, in case you are wondering what I am referring to). I think it is safe to assume this is an F dominant seventh (f-a-c-e♭). (I don't know the names for chords with sevenths in English.) To determine which inversion a chord is, one should always look at the lowest note, so here it would be first inversion.
How I identified the chord: I recognised F-A-C as an F Major chord. But even if we ignore that, when going up from A (A-C-E♭-F), C (C-E♭-F-A) or E♭ (E♭-F-A-C) we get a major second every time. There are not so many chords with major seconds, so I think the safest assumption is the F dominant seventh. I personally do associate all above inversions with this chord.
As for the lowest note: I already argued this is an F chord. So the root chord would look like this, from low to high (obviously, the root note comes first in the root chord): F-A-C-E♭. Then we just count these notes, starting at zero:
F-A-C-E♭ 0 1 2 3
And we see A maps to 1 => first inversion. I think this rule that the lowest note determines which inversion a chord is is somewhat counterintuitive, but that is how it is. (You need some rule as soon as there are octaves (or huge intervals) in the chord to be identified.)
Your example is an F dominant seventh in 6/5 position (first inversion).
Steps for determining the inversion of a chord: