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.

The problem happens when you change the syntax definition of an already saved file. This bug will be fixed in the next update.

The workaround is to immediately change the syntax definition after you create a new file and before you enter any characters.