So I understand chord inversions, but my question is this, do the notes have to remain in relative order? with triads there are only 3 possible inversions and because there are only 3 notes you're only shifting which note goes in the base, but what about something like a Cmaj7add9? I understand inverting it like this: d c e g b ect. Ect. but what about if I'm chording on the guitar and the note order is changed completely? Random example: d e b c g. Is that still the same chord? Or do the notes have to follow this "c e g b d" order to remain a Cmaj7Add9?
Cmaj9 (no need to 'add' 9, it's a natural order) has the notes C E G B D. When played using this order, it sounds at its best. Root at the bottom, extensions spreading out uniformly, every note a third away from its next door neighbour.
Even keeping to root version, there are 24 voicings available. Changing the root note allows 120, if my sums work.And that's in close position, Some of these sound good, some awful, due to neighbouring notes not getting on well. Try (on piano, it's pretty well impossible on guitar) playing the chord as B C D E G . The interaction between the lowest 4 notes doesn't do it any favours.That's in close position. In open position, most inversions will sound acceptable, but that's a non-starter on a guitar.
But, yes, those notes, played simultaneously, will always constitute Cmaj9. You specify guitar, which throws up another problem. In order to play all of those notes, they must fall under the span of your fretting hand. Many of the voicings are impossible to fret, and there aren't many compromises. There's also one string which could double one of the notes, or be muted, as happens a lot with more complex chords.
So, in summary, the version which grew from C in thirds is probably (possibly?) the most convincing, but any inversion is still Cmaj9. But, a lot of them won't sound so good, and most of them won't be playable on guitar.
There are 6 voicings of the triads, 2 of each inversion. For example (1, 3, 5) is the root inversion but you can also have (1, 5, 3). Similarly for the others. They are the same chord but each voicing has a unique character and in my opinion can be though of as different from a harmonic perspective. Choice of inversion is somewhat driven by the other chords in a progression and how they harmonize a melody as they may have a "strong tendency" to lead to a voicing of another chord in the same key. Like moving from V7 to I. They all have that classic dim 5 resolving to a maj 3 but some combinations of chord voicing will sound stronger than others.
As for seventh chords and beyond, yes you can have inversions. In fact there are 4 inversions of the dom 7 chord and they are described in any music theory text. The 3rd inversion has the 7th in the bass. To your other question, yes. Any reordering of the notes represents the same chord. Not all ordering will sound nice but they all represent the same chord. On the guitar we frequently pick whatever ordering we can play.
First things first: The chord name will always be influenced by the context it is played in. There are plenty of examples of this, like diminished 7th chords and their 4 possible roots, but I think that the question here is actually more "is this still a valid voicing of the chord".
Secondly, I'd call that chord you listed Cmaj9, and most musicians agree with me, because the "add" is required only when there's a break in the order of extensions (Cmaj9 implies that the 7th is also present, and what's more, major). We don't call C-E-G-B♭ "C add dominant 7th" because it's redundant.
To get to the point, the order of the notes does not matter. Any combination of C, E, G, B, and D can be a Cmaj9 should you so desire. However, there's a subtle difference between inversions in that they contain different intervals and therefore a different feel. B-C-D-E-G is a valid voicing of the chord, for example, but it sounds way different than C-G-D-B-E. And it's not only about the distance, either. Try B-G-C-D-E and you'll see what I mean.
Related: How does one invert chords with a range larger than an octave? Is the 1st inversion of G9 B-D-F-G-A? Often times, the inversions of chords are quoted as "take the bottom note and move it up an octave". However, this falls apart with 9th chords. I think it's better to think about inversions of chords in this way: "Start at the bottom of a chord, then move every note upwards to the next highest chord tone". This works with guitar chords especially well too, as one can preserve the general sound of each set of inversions. In this way, G9's 1st inversion is actually A-D-F-G-B, preserving that major 9th interval!