I've had a look at a similar problem yesterday. I'm not sure if this is even possible for a sandboxed Mac App Store app which typically can't launch other arbitrary apps or execute AppleScript.

You can just use File -> New or Cmd-N and the Open dialog will close. 

Which part didn't work? You should be able to paste text in Textastic by doing the following: tap on the code editor once to bring up the keyboard. Then tap again to bring up the context menu where you can select "Paste".

Because that is the standard behavior since Mac OS X Lion.

You can change this behavior in system settings:

Will be available in the next update.