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I use the official Dropbox iOS SDK: http://www.dropbox.com/developers/reference/sdk

It does use the REST API.
This if fixed for the next update.
Looks like a bug. As soon as you hide the keyboard, the behavior is back to normal for me.
See my answer below on when https is used.
Error -1004 is "Cannot Connect to Host". Please check your internet connection - the Dropbox server doesn't seem to be reachable.
I use the official Dropbox API. It uses HTTPS for things like login and downloading files. It uses HTTP for less critical things like loading metadata and sending commands to create folders or move files. I don't intend to change that.
You can try to use the TextMate AsciiDoc bundle to get highlighting for it:

Have a look at the the manual.
From the article: "Enabling the remote inspector on a UIWebView in an app running on the device doesn’t seem to work."
Wow, this looks really horrible to be honest :)
It will be in the version after the current 3.2.