: How to determine which inversion a chord is? Say I'm given the notes like this. How do I determine whether it's already in root, 1st, 3rd, or 7th inversion?

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This is an F dominant seventh chord. The inversion is determined entirely by 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. A chord, by definition, is a stack of thirds, so you have to reorder them until you get rid of the second. (Exception: incomplete chords skip thirds.)

To find out the inversion, we just have to list the notes in the order in which they are all a third apart: 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 for first inversion.

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Your example is an F dominant seventh in 6/5 position (first inversion).

Steps for determining the inversion of a chord:

Use your ear to try to detect a particular character (is it major? minor? Tristan chord?)
Write down (or think of) all possible voicings. It often helps to pack the notes into as small an ambitus as possible, and/or to arrange it as a stack of thirds.
Does one of the voicings match the character you detected in step 1? If so, that's the root position, now figure out the inversion (determined solely by which is the lowest note).
If you have multiple good answers or can't determine the character by ear, examine the harmonic context. (Is the chord a massive, imposing feature of the music, or simply a quick way to get from one place to another? The latter type may not have well-defined inversions, as it may contain suspensions or chromatic alterations.) Also consider the spelling of the chord. (In your example, if you had E# instead of F - weird, I know - you'd have Adim(#12) in root position rather than F dominant seventh in first inversion. This issue becomes a lot more relevant when dealing with, for instance, fully diminished seventh chords, which have several equivalent spellings. A convenient tool here is to simply remove all accidentals, changing the notes to naturals, and see if you get a chord you understand.)
If all else fails, try to find any triad within the notes (if multiple triads, the one that sounds the most like the original) and base your analysis on that triad, calling everything else an extraneous addition.