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I was taking a look at this chord progression and attempted to play it just using the roman numerals.

Seeing the V(G) I played what would be the right most G in the image with the B and D above it so that all the notes are in one octave worth of the scale (Is there a better way to describe this?) but clicking on it shows the exact keys played are different.

Is there any possible way to have known that the G would be above the B and D or is that information not part of the roman numerals above and simply a style choice / inconsequential.

Where you play the notes - in which ocatve, or even how they split - root in bass and others in r.h., or split them two with each, etc., is very much up to you, as the player. It's called voicing, and as PiedPiper says, when all the notes are as close as possible, it's called 'close position' or 'close voicing'. The opposite being 'open voicing'.

It matters not how many of each note gets played - and sometimes the 5 gets omitted, as the root gives the chord's name, and the 3 gives it its major or minor, an important fact when playing/naming chords. There really is no right or wrong with voicings. There is of course 'appropriate'!

With RN, it merely gives the number of the chord in that situation. There are so many different voicings available, it would be impossible to use a simple means to tell. Except, of course, writing the actual notes. But then that wouldn't be immediately transferrable to different keys.

The only thing about the distribution of voices that Roman numeral analysis provides is which voice is in the bass.

Just some background on the symbols...

The Arabic numerals added to Roman numerals are a historical hold over from figured bass and an approach to harmony that was more about counterpoint. The figures told a keyboard accompanist what type of chord to use to harmonize a written bass part, but they were free to improvise the voicing and figuration of the chord. Most of the time there would be written melody parts as well. So in essence the Arabic numerals told the keyboard accompanist how to play harmonic filler with their right hand with no particular concern for the exact voicing or figuration of the chords.

So there is a odd historic evolution of the chord symbols. Figured bass put numeric figures on a written bass part. Later in the Roman numeral analysis (RNA) system the figures were added to the Roman numerals - which give only a chord root - and the bass part is abstracted. Neither system gives specifics about the upper voices.

EDIT

After re-reading your question I'm confused about what exactly you are trying to figure out.

Is there any possible way to have known that the G would be above the B and D...

In Roman numeral analysis (RNA), yes, you can indicate the G is above the B and D by using the 6 figure - like G: I6. But, properly speaking that I6 means the third of the chord is in the bass. It does not necessarily mean G is above the D. Listing tones from bass up to treble I6 can mean B G D or B D G.

...or is that information not part of the roman numerals above...

It isn't clear if the screen shot you provided is of software that cannot display the Arabic figures on the Roman numerals

...or is that information... simply a style choice / inconsequential.

I would say it is not inconsequential and important to style. Pop/rock style doesn't typically use RNA for chord symbols so it's a bit hard to explain. Pop/rock often uses chord names and 'slash notation' - like F/C to mean an F major chord over a C bass. In RNA it could be C: IV6/4.

Pop/rock often uses chords in root position in which case the Arabic numeral figures would see inconsequential. Or, if the slash notation is used, some people aren't even aware of the concept of chord inversion.

Nevertheless, chord inversion is important in rock music. In a progression that is supposed to be played with all root position chords, if the bass starts playing the thirds of the chords, it won't sound right. Regardless of the type of chord symbols used, any software's ability to display a chord symbol, or a musicians understanding of these things, the inversions are being used and they are an important aspect of any musical style.

The roman numerals say nothing about how you distribute the notes in the chord. The V chord contains the notes G, B and D, but you can play them anywhere.

If you play the chord so that all the notes are within one octave, this is known as 'close position'.