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Hello,

it's possible to use Firebug lite in Textastic: http://getfirebug.com/firebuglite

The problem is, that its user interface doesn't work particularly well on a touch screen. But it's definitely usable.
Just tap on the blue arrow on the right side of a connection to edit the FTP settings.
Tap on the editor with two fingers to bring up the cursor navigation wheel.

See this entry in the manual: http://www.textasticapp.com/v3/manual/lessons/How_can_I_bring_up_the_cursor_navigation_wheel.html
That's only possible if you upload it to a remote server.

But the rendering in Textastic should be the same as in Safari as it uses UIWebView which is essentially the same WebKit engine. Though here might be differences between accessing local and remote files due to things like cross-site-restrictions.
Just use the code/preview button in Textastic to preview local html files right within the app.

To open the file in Safari, you would have to upload it to a webserver and enter the URL in Safari. Safari can't access local files that are located in other apps.
You can preview an image file by tapping and holding it and choosing "Preview".

You can switch to recently opened files by tapping on the current file name in the toolbar on the top.

You can create folders to manage your projects.
Sounds like a good idea.

Did you know that you can display the image by tapping and holding it and then choosing "Preview" from the menu that comes up?
Yes, these are all things that are on my todo list :)
Just create a folder before you download and then download the files into that folder by selecting it on the left in the File Transfer screen. 

There's no need to download it to the root folder first and moving it afterwards unless I'm missing something.

Moving files is implemented similar to how you can move mails in Apple's Mail app.