does anyone know how the whole "emos cut themselves" stereotype got started?

It came from the stereotype that all emos are depressed. Emo is just short for emotional, and that doesn't always mean depressed.

But since they're seen as depressed, people think of them as wanting to die, so they hurt themselves.

I always thought that being goth was just the style of clothing, and that emo was the type of music and poetry and stuff. But don't take my word for it, I'm really not sure...

haven't got a clue bout the emo stereotype but the difference between goths and emos is that they dress differently and they into different music as well

Well, i'll answer the first one (because the others have already be answered).
I have no idea really how it started. Thats stereotypes for you. There Stupid and pointless.
Self harming has been around for a long time - even before 'emo'. It may have be passed down,as a way of identifying emo,maybe? My thinking is that people self harm to relieve the pain of something,because the physical pain is easier to live with than emotional pain for the rest of there lives. I think only posers do that to get into the emo culture or for attention. Cause it's not all nabout being depressed all the time or cutting - as they say. Thats how i see it,from my experience.