soundcloud API - JSON track count does not match profile track count
To answer your question, it definitely seems that the SoundCloud API returns all tracks from the artist, no matter what state those tracks are in (public, private, etc.).
As far as the rest goes, Sound Cloud is notorious for being strict with how artists' content is handled in terms of embedding or having it used on other sites. It is usually up to the artist from my understanding, but in general, they want artists' content to be protected, especially if they are under a contract with a record company or something.
As for your web app, you'll either have to work around it or just ditch it completely, unless you can create some sort of scraper that can get more relevant info. If your site wants to play music directly from the site, however, you may be at a loss.