Your comments

Yes, you can upload any type of file. 

If you want to know how to get pictures into Textastic from the Photos app, have a look at the comments in http://feedback.textasticapp.com/topic/9278-import-pictures-from-the-photos-app/ to see how you can do this with the Dropbox app.
How did you create the symlink? Do you have a jailbroken device?
If you have picture downloaded to your local file tree, you can tap-and-hold the file and select "Preview" to view the picture or tap on the button on the right side of each file in the file transfer screen and choose "Preview".
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.