I have a 7th chord: A-C#-E-G , in the key of F# minor and would like to symbolize it using the Functional Chord symbol: ( I 7 / III ). Would it be correct to identify it this way ? Using the root quality chord symbol I have simply named it A7 . So can Functional Chord symbols include I7/III or the triad I / III ?
Based on a follow-up comment to the original question, the intended answer is almost certainly V65/VI. The textbook asked to identify the applied chord, and at this point, the only applied chords will be a V(7)/x or viio(7)/x. Later in your studies you may encounter IV/x, or even what we call extended tonicizations, which I discuss here.
As such, this A7 chord (with C♯ in the bass), which is dominant of D, would be V65/VI, because D is VI in the key of F♯ minor. (Similarly, it would be V65/IV in A major).
I would advise against writing iii7, which would suggest a G♯ as opposed to a G♮ in the chord.
It's not needed. III7 is sufficient. Normally, applied chords are V/ or vii0/ or maybe IV/ or a pair like ii-V/. The point is that one is locally adding a V chord to tonicize another chord. Common (in a major) is V7/V7 (D7 in C). In minor, V7/V does occur but more common is V7/III (Bb7 in C major). Unless acting as a temporary dominant (or maybe subdominant), there no need for the / notation. I should point out that the sequence (in C) of C7-F is often written V7/IV. It was common in 1930s popular music: C7-Fm-D7-G7 was a common turnaround and analyzed as V7/iv-iv-V7/V-V7.