Suppose I'm playing a C major triad, does the order matter in regards to pitch?
Does the C have to the be lowest note?
Does the E have to be lower than the G?
Are the rules the same regarding other types of chords?
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Order of the notes in a chord will not change the name of the chord or the quality of the chord (major, minor, dim, etc).
However it depends on what you mean by "does it matter".
Sure it matters for voice leading, stylistic tendencies, and should be considered when writing a piece but if you need to play a C major triad it will be a triad regardless of the lowest note. FYI a chord that has a note that is not the root as its lowest note is an inversion of that chord. So an E in the bass of a C major triad is said to be in first inversion.
One note is when a composer (sheet music, lead sheet, etc) specifically tells you to play a chord over a certain note. For example if the chord is C/G you should play a voicing of a C major triad that has G in the bass (lowest note) which is the second inversion and could be called "C over G".
In terms of what you hear as a whole when listening the order of the notes does not matter. But having different parts of the chord as the bass notes, say the fifth or the third or the major/minor seventh, which are all common to use, are what are called inversions. I won't go in to all of the various ways you can use inversions because there are too many to talk about but the most common one would be for planning out a descending/ascending bass line. This is incredibly common on jazz arrangements. You should think of them in the sense of achieving different "lines" or riffs that get you from chord X to chord Y -- inversions just make it easier to do this as you then are not limited to only following the root notes of the chord structure. which would be incredibly boring and lackluster.
In some variants of CPP harmony, the order of notes in a chord does matter. Baroque (and Rennaissence theorists (those living in the 1500-1750 or so period, not moderns discussing old music) did consider the inversions of chords as being different to a greater degree than most pop or jazz theorists. Things like a 5-6 technique (moving the harmony from C-E-G to C-E-A) were taken as closer than C-E-G to E-G-C as the bass note is the same. This was done even though inversions were known. This idea does fit well with a figured bass approach also.
Another example is the 6-4 chord. A cadential 6-4 chord occurs in progressions like G-C-E-G followed by G-B-D-F then some form of C chord. One can read the 6-4 as being a version of a C chord or as a decoration of a G chord. (I prefer to think of them as a object in themselves with various types of uses.)
It's not wrong to consider all chord inversions as being variants of the same chord; but it may be helpful to consider other ideas in addition. Similarly one can treat vii0 or vii07 as a V or V7 with the bass elided as well as a chord based on scale step 7.(Minor key harmony has a diminished chord on step 2 also which seems to be generally based on that step.)