The I,III don't have their 7th lowered. The scale you are looking at is most likely the G natural minor scale and not the G Harmonic minor scale. The notes that are included in the G natural minor scale are the same that consist the Bb Major scale (Bb, C, D, Eb, F, G, A). The correct thing to say here is that on the V and the VII, some notes are raised. These two chords are called Dominant Chords.
What you need to understand is the role of the Dominant chord. This chord is the V degree of your scale (and can also be substituted by the VII) and it
has the role of creating instability that requires the tonic for resolution.
So, you see, point of a dominant chord is to lead the chord progression to the first degree of you tonality (in your case G minor).
When you add F# to a chord, it becomes the leading tone of your scale. The leading tone is a note that leads to the tonic. So, in your example, if you have a F#, it has to be resolved to the tonic -> G.
To make it even more simple: In a chord progression, after you play V (or VII), most likely, the next chord will be I. (this being the point of the dominant chords). Thus, you need the leading tone to make it clear to the ear of the listener.
The rest of the chords (II,III,IV,VI) don't necessarily preceed I, so there is no point to include the leading tone.
Quoting Wikipedia on Leading Tone:
Melodically strong affinity for and leads to tonic.
Of course, there are exceptions for all the -rules- I mentioned above; I just tried to mention the really basic stuff, so you can understand it.