Let's say I have an existing page, and I want to use Google Data Highlighter.
Later on I got access to its back end. Can I put Schema.org code directly to that same page?
You can, though assuming they're both marking the same content you'd probably want to turn off the Data Highlighting when you've added the code to avoid any possible conflicts.
On the other hand, if you're using Data Highlighting for one data type and actual code for another, I see no reason why the two approaches couldn't be used concurrently.
Why would you? Schema.org markup can be consumed by all major search engines and other user agents, too.
Google Data Highlighter is (as far as I know) a proprietary system that has no benefit beyond Google. The highlighter is better than nothing but if you can handle structured data, I encourage you to do so.
Late last week I completed a page a fair amount of structured data (which is not linked to other pages on the site) and submitted them to be crawled by both Google and Bing webmaster tools...Bing is the only one that's sent me organic traffic thus far.