i just started learning about diminished and augmented chords, and i need some clarification. for example:

this could be C#(or Db, depending on context) dimished as it contains the root, minor third, and diminished fifth, correct?

|------|
|--5--|
|--4--|
|--5--|
|--4--|
|------|

and this could be an A augmented chord, as it is comprised of the root, major third, and raised fifth, correct?

|-----|
|-----|
|-----|
|--3--|
|--4--|
|--5--|

now i just have one question: does the order of the notes in the chord have to be mandatory? that is to say, could i have the augmented A chord look like this?

|-----|
|--2--|
|--x--|
|--3--|
|--0--|
|------|

it has the same notes, but instead of having the raised fifth on top, i flipped the notes around and put the major third on top. is this ok?
The notes can be in any order. But your C#dim also has a minor seventh, so it's a diminished 7 chord.
The first is a Db m7b5 chord (some call them half-diminished). It has a b7; a full dim7 chord has a bb7. Still a beautiful chord, but a different function, and not symmetrical like a dim7. Both chords are built from a diminished triad so you're on the right track.

That is indeed an Aaug chord. Or a Dbaug. Or an Faug. Symmetrical, right.

Augmented triads are super cool. Why? Lower any note in the triad and you have one of three different major triads. Raise any note and you have one of three different minor triads. This pattern is just ripe to be exploited: It can provide a very solid anchor around the fretboard to find chords quickly, because there are only 4 possible augmented triads. Pat Martino has written a lot about this.

There are rules that apply to voicing, and by rules, as always, in music they're guidelines. For now, voice chords in a way that sounds nice to you. This on top, that on top, on the bottom, leave out notes, whatever.
oh right, i didn't realize about the m7 in the diminished chord. thanks for the help though.