For instance, if I have a url for a website that uses Disqus for managing comments, is there anyway I can access the comments on that website via some API call? As far as I know, the API calls only enable access to comments on applications owned by you...
As long as the comment is approved and visible on the site, it's available in the public API. You would need to know the disqus shortname of the site, which you can usually find in the page source by locating the var disqus_shortname=... variable.
If you want to modify anything about those comments, that's where authentication and permissions comes in, but it doesn't sound like that's part of your question.