Okay, this is four semesters of music theory classes, 4 semesters of piano class and a semester of guitar along with a lot of self-guitar-teaching, and about 7 years of songwriting talking here, but I am very crappy at explaining things (One of the reasons I’m not a Music Ed. major… ) so I’m not sure how clear I’ll come across.
A) It varies. It depends on the writer, really, and the song itself even. It also depends on if you’re talking harmony as in the accompaniment to the melody, like guitar chords or piano, or if you’re talking harmony as in companion to the melody, like backup vocals. I’ll go with the first sense first, since that’s what I think you mean.
Before I learned guitar or learned any music theory, all my songs were just a melody and lyrics. I go back now, and it’s occasionally pretty difficult to just add guitar or piano to those old songs. (although my still-mediocre guitar playing may have something to do with that…) So, depending on the melody itself, it can be hard to come up with a pleasant-sounding accompaniment. Now, I write all my songs with my guitar, and the process varies. Most often I come up with a snatch of melody and a lyric, and find chords to go with it and build from there. Sometimes I find a chord progression that I love, and build a part of a melody that sounds good with it, then add lyrics and build it along that way. This varies among songwriters, really – alot of it depends on their main musical focus. I’m a singer, and a writer, so the words and the melody are the most important part to me, so that’s usually what I build from; although the guitar or a piano are now pretty much always an integral part of the process, the spark is usually a lyric or a bit of a tune. On the other hand, I have a couple friends who are guitarists - their songs often have no real lyrics or even solid melodies, it’s progressions, riffs, etc. I’m honestly not sure how to explain it, it’s really different with each writer.
If, on the other hand you mean harmony as in back-up vocals and such (I’m thinking barbershop quartet stuff and the like), complements to the melody, then I would say that almost always comes after the melody. I know I never come up with background vocals for a song until I record the instrumental and lead vocals and decide it needs something more. I’m not sure if this is the explanation you’re looking for… but I did my best.
B) As for guitar chords being the left-hand of the piano, that also varies according to conventions and whoever transcribes everything. The notes of the guitar chords will probably be there, but not necessarily in the left hand, depending on the style. I know in my last piano class they told us a couple different things – either the chords are entirely in the left hand, usually in whatever inversion is easiest for reach, and the melody is in the right, OR the left hand plays the bass notes of the chords, often doubled in octaves, while the right hand plays the chords and the melody (since the notes of the melody are almost always part of the chords with the addition of various non-chord tones for connection or embellishment, this isn’t hard). And then there’s the option to play half the notes in one hand and half of them in the other, split the difference, if you will. Once again, it really varies.
Am I any help at all? I don’t know. I’m bad at explanations, I think. But I did my best, short of lending you my theory textbook.