No, a #3 is a 4 (or in a 7th chord, 11, or when the 3rd is replaced by the 4th, sus4). In this case, the 3rd is raised not a half step but a whole step, to #4 [#11] or b5 (but the latter isn't viable here). The proper name would be C7#11. No one following conventions would use C7x3 (i.e. 3 double-sharp), while C7#3 is not only contrary to conventions but simply wrong.
John also misjudged an interval in guessing Am7b9#5, because the 5th of Am7 is raised a whole step, becoming a 6th or 13th; the correct name would be Am13b9.
When spelling chords, the convention is to go from 4th string to 1st (just like we read across grid diagrams left-to-right), not from 1st string to 4th: thus 0655, not 5560. Also, for open strings, write o or 0 (lowercase oh or zero); X or x means to mute or skip the string.
I also suspect your GM7sus2 is just D/G. If it truly is the rather oddball GM7sus2, might a plain GM7 (0675) be more effective?
I think in this progression your mystery chord 0021 is D+7/G rather than C7#11. Basically, you're using the G note as a ground or drone, so consider the chords without regard to the G, then add the G when necessary using slash notation:
D/G - C - D+7/G - D - Gm
Given the voice leading, I'd actually name the mystery chord D7b13/G, with the b13 [b6] leading downward to the 5th in the following D chord: compare to this D7b13: 2021. It's also possible that John's guess of an altered Am7 chord (Am13b9) is right here, making the final progression ii-V-i: a standard major mode progression with a surprise shift to minor on the last chord. If the melody or bass is doing something other than sitting on the tonic note, that might decide the matter, but with only these contextual clues, it remains debatable.
By the way, my guess (particularly given the last chord) is that your key is G minor rather than its relative major key, Bb. Bb would then be the III chord—far more common to omit the III chord in a song than to omit the tonic (I) chord. Although the final progression more fits G major than G minor, such "parallel" mode shifts are a common device, and the main mode is reestablished on the final chord.