Sure, chords can contain major and minor thirds at the same time! We see 'em all the time in jazz. I call it a major-minor chord (which however is quite different from a major-minor seventh chord). It's also called a mixed-third chord. In jazz chords you will also sometimes see it notated as a ♯9 instead of as a ♭3, though that implies a slightly different use for the added note – that is, as part of the harmonic "juice" – i.e. a triadic extension to the chord – instead of as the bass.
In theory, every note in a tonal system can be represented by an interval relative to a given root. For slash chords, you use the root of the chord above as your root. In this case, ♯2 (♯9) and ♭3 are usually considered enharmonic, and therefore either could be used to describe G, so you have to resolve the ambiguity. You use the note name as your guide. If your root is some sort of E (e.g. E♭, E, or E♯), and the note is some sort of G, its interval is some sort of 3 (think of G as two "note names" away from E). If the note is some sort of F, its interval is some sort of 2. So in this case, G is considered a ♭3. (A ♯2 in this case should be notated F double-sharp).
In practice, however, there's not a single correct answer to "how to think about the G". In fact, when a slash chord gives a bass note that isn't related to the chord, I think it's often more illuminating to simply think of the bass note as a note or part that is independent of the chord above it, rather than trying to think up a warped chord description that includes it. I suspect that might be the case here.
It may be that the music is polytonal; i.e. one part plays in one key while another part plays in another. In that case, trying to mash the two into a single functional harmonic structure and understand the piece that way is likely going to result in chaos and tears.
It may also be that the bass note is not a functional part of the harmony, but rather a line that moves linearly and logically through a progression, but in harmonic conflict with the harmony above it. In this case, trying to figure out the note's position in the harmony might bury its true significance as part of a horizontal line. This kind of horizontal thinking is especially important to develop as a bass player, and I think is the key reason slash chords are written the way they are, instead of via intervals: so that the progression of the bass line is crystal-clear regardless of what comes above it.
The language of chords and harmony has developed not just to tell you what notes to play, but why. It's a powerful tool for communicating a lot of musical information quickly, but it's not the only tool at your disposal.
P.S. Interestingly, in 20th century theory, when wrangling with atonal music, musicologists have given up on the traditional interval system and switched over to using a purely numeric interval system based on half-steps. In that system, let E=0. Then G=3 (it is 3 half-steps away from E), G♯=4, and B=7, so you might call the chord . I've only seen that done in atonal music, though.