Q1) What is the definition of key?
My grandfather's music dictionary says:
key: A scale, or series of notes progressing diatonically.

If I write a song that uses the notes of the E minor harmonic scale exclusively, without falling back into an E natural minor or some other related diatonic scale:

Q2) Can the notes that I am using be said to be "natural" to the E h.m. scale?
I guess you could say that.

Q3) Can the song be said to be written in the key of E h.m. ?
Definitely.

Q4) Can the key signature relfect the fact that the only notes in the song are those belonging to the key of E h.m. ?
No. Key signatures (as far as I know) don't reflect harmonic minor.

Q4) If the signature can't indicate minor harmonic tonality, what are the conventions for marking the key signature and notating accidentals?
You probably would use the E minor/Db major key signature, and mark the non-natural tones with a flat symbol.


Hope that helps.