WordPress : using AJAX to get posts & Sidebar Content to an external application

If it is all on same server it is rather easy to load WP engine and parts you need, see Integrating WordPress with Your Website.

Otherwise there is probably better to get specific data (rather than whole page) from WordPress remotely, but there are a lot of approaches to that. Also I am not aware of any off-the-shelf code that allows to do that for sidebars. For post content there is XML-RPC support and plugins for other formats (I think I read about JSON one).