Are there any websites were an amateur writer can post a story or parts of a story and for a fee or for free the writing will be posted for a 'while'? (And maybe the story will be 'noticed' by a publisher.)
I'm a fan of http://www.writersbeat.com myself. The site itself is a forum, and you'll be able to publish parts or complete stories while getting loads of great feedback. A very helpful and friendly community.
I don't know if there are actual agents or publishers lurking there, but it's quite large so I would not be surprised.
OpenBook.pub as Ivo suggested is also a nice site.