As a fanfiction writer myself, too, I have to say that I don't think that writing fanfiction 'is cheating', or 'doesn't count as real writing'. I think that fanfiction can be a good way to develop your writing skills, but shouldn't be the only thing you write, unless you're not really serious about writing... Although I'm not saying that everyone who only writes fanfiction aren't serious writers. I haven't used songs in my writing before, althought I have planned a couple of songfics, but it can be effective if written well.
Although I'm all for fanfiction, I have to agree that it has a bad side - in some cases, a VERY bad side. Fanfiction is often full of clichés, bad spelling and grammar - Basically bad writing in general. Then you get some stories that are worth reading, but nothing special. If you're lucky, every now and then you come across a really good story, which makes wading through the many bad ones worth it.