Facebook comment is posted on personal page, not Application

When check box "Post on Facebook"is clicked, the comment is posted on user's page (who made a comment) and on my personal page - but not the FB application page.

Yes, by default the comment box widget is tied to a specific website URL/application, not a Facebook Page. That checkbox allows the post to show up on the user's profile who made the comment. If you'd like to tie your commenting on your website to your Facebook page, then you will need to look into the Comment Mirroring capabilities of the FB comments widget. The Facebook documentation describes how to set this up.

So in the Social Stream module there are the posts from the app, so it means that the Facebook channel is configured correctly. How I can get comments from my visitors on this page as well?

In the SCSocialStream widget, make sure the "Show Comments" option is enabled. Once Comment Mirroring described above is set up correctly and the comments are showing on the FB Page, they should then be displayed in the widget as well.

-Melissa