No, currently this is not supported, sorry. Previously this wasn't allowed by Apple at all, but they recently made some changes to the App Store guidelines which I'll look into.

You can add custom syntax definitions to Textastic by using TextMate bundles. See
http://www.textasticapp.com/v6/manual/lessons/How_can_I_add_my_own_syntax_definitions__themes_and_templates.html

A quick search for "Kotlin tmbundle" in Google brought up this TextMate bundle which should work:

https://github.com/sargunster/kotlin-textmate-bundle

Ok, I just saw that on macOS, the code completion window is not visible if the window is maximized and you are editing in the last line. That's definitely a bug. I've added it to my todo list.

Currently it is not possible, but I'll consider adding some extra space after the last line so you can scroll down further.

I'd love that, too, but as far as I know this is not possible at the moment. 

The launch storyboard does not run code and so it cannot access the theme settings and act accordingly.

Thanks for the screenshots! I'll look into it.

Did you see that you can enable the standard copy/paste buttons above the keyboard in Textastic's settings? The setting is called "Shortcuts".

They are disabled by default to make room for Textastic's swipeable additional keys.

You can already set a path for (S)FTP connections. Dropbox always starts at the root folder at the moment.

Just do a single tap on the code editor to make the keyboard appear. You can then select text by double-tapping.

If you do a double tap instead when the keyboard is not shown yet, you only select the word under your finger. This allows you to select text and copy it to the clipboard without bringing up the keyboard.