In root position a B7 chord in the key of E minor would be B,D#,F#,A and be made up of scale degrees ^5 ^#7 ^2 and ^4. So in 1st inversion the bass would be D#. If I was to add ^3 as a soprano note over this what chord options could I have? I know in root position ^3 is an extension of dominant but how does this fit in over D# in the bass? Are there other chord options with D# in the bass and G in the soprano? Is what I am hearing an inverted augmented chord (^#7 ^3 ^5)? Is this even a real chord? I quite like the sound of that resolving to a proper V65 chord because the ^3 resolves to ^2 which then resolves to ^1 in the tonic chord. So I kind of see why I like the voice leading but not sure what chord this could be.
It is an augmented chord. As you're probably aware, rather like diminished chords, which could have basically 4 names, augmented chords could have basically 3.
Dependant on key. So, in key Em, D♯+ or G+ aren't as appropriate as B+, but it may make a difference as to what follows it. Generally speaking an augmented chord precedes one a fourth higher. So B+ > Em sounds good. How it gets voiced is basically up to the writer/player, but a common move would be D♯>E, the oft-used one semitone shift. In this case, the only note needing to move.
Isn't it intuitive - if you have a dominant chord built on the leading tone in a minor key with ^3 on top, you have a V13 chord in first inversion!
At this point, ditching the 5th of that inverted V13 chord so you have (drawing from your example in your question) D♯-B-A-G instead of the clash-producing D♯-F♯-B-A-G is a good idea.
An (inverted) augmented chord, D♯-B-G, is indeed in that V13 chord in first inversion, but hopefully, the 11th in that chord makes the augmented chord less noticeable.