Well, when one wants an exhaustive list of obscure exceptions to every conceivable musical principle, often the person to turn to is the great Ebenezer Prout, whose books are full of just that sort of stuff. Prout doesn't disappoint in his book on Fugue, where he notes:
Though there are limitations as to the note of the scale on which a
fugue subject should end, there are none as to that on which it should
begin. In an enormous majority of cases, the subject begins on either
the tonic or dominant; but numerous examples are to be met with of the
employment of the other degrees of the scale for the initial note. We
give a few instances of each...
Here Prout notes that subjects beginning on the supertonic note are "rather rare" while those beginning on the mediant "are also not very often met with." The two examples he provides for the mediant are from Beethoven's Missa Solemnis and from Cherubini's 4th Mass:
I don't think fugue subjects beginning on a mediant are a large enough group to generalize about their treatment. But yes, there is no reason you can't begin on the mediant, just as composers began fugue subjects on every other note of the scale (see Prout link above; also, another answer mentions a WTCII fugue that begins on the leading tone, while another fugue in WTCII begins on the supertonic).
The standard notes are tonic and dominant for the obvious reasons that they work best within the normal world of fugue subjects and answers that generally have strong tonic/dominant relationships. Also, most (though certainly not all) fugue subjects begin by strongly identifying their key, and tonic/dominant notes tend to do that the best. Beginning elsewhere may introduce a few complexities in counterpoint and harmonization, but nothing that a skilled fugue composer wouldn't be able to deal with.