Yes, any seventh chord has inversions. No matter what chord quality, the seventh chords are inverted the same (actually, all chord qualities are inverted the same way).
It's possible to invert ninth chords (and beyond), you just have to reconsider the method you use to produce inversions. The way most people learn inversions is by taking the bottom note of the chord and moving it up by an octave. This works fine, until your chords get larger than an octave. Doing that process to a root-position stack-of-thirds C9 chord results in [E G B♭ C D]. This is not the same kind of voicing as the root-position one, and that's because this chord is voiced entirely within an octave, while the root position one had a ninth interval. The two aren't really equivalent.
The better way to conceptualise inversions is by moving every note of the original chord up to the nearest chord tone. Try it on an A major chord: A moves up to C♯, C♯ is moved up to E, and E is moved up to A, making [C♯ E A]. This is exactly what the regular method produces, which is good. This works the same for seventh chords as well.
Now, do the new algorithm on the C9 chord: C moves up to D (D is a chord tone, remember?), E moves up to G, G moves up to B♭, B♭ moves up to C, and D moves up to E. This means the first inversion of C9 should be [D G B♭ C E]. Think about that: D as the bass note makes perfect sense, as the bass note should ascend as the chord is repeatedly inverted until it gets back to root position. The original chord had a range of a ninth, and so does the inversion. And when this process is repeated, it cycles back to itself using four different inversions (the fifth is root position)!
I think the best evidence for this improved algorithm, though, is listening to the inversions you get from it and comparing it to the original chord. No matter how many notes are in the chord or even how far apart the notes are (try inverting [C G E] to [E C G]), the inversions just sound like inversions of one another in a way that the algorithm most students are taught simply fails to emulate (because the spacing of the notes is preserved in the better algorithm).
Finally, I mentioned inverting chords with more than 5 notes. Technically, it's possible to invert 13th chords, but because 13th chords contain all the notes of a scale (in theory), any inversion of a 13th chord sounds like a different chord entirely (Cmaj13 in first inversion would be identical to a Dm13 chord - try it out, if you don't see it immediately). At a point, one has to ask what the musical value of inverting enormous tertian chords is, so to summarise:
To invert chords, move every note upwards to the nearest chord tone. This is cool because it allows the voicing and intervallic properties of a chord to be preserved during the inversion, no matter how the chord tones are spaced in the original voicing.
One corner case I forgot about: symmetrical chords. You mentioned diminished 7th chords, which are symmetrical. Because the interval pattern to create the chord repeats, every inversion of a diminished 7th chord creates a new root position diminished 7th chord on a new root. Same for other symmetric chords like augmented chords. One could argue that those chords have inversions, but they just sound (and look, for the most part) the same as other root-position chords.