Duplicate selected text to a new line VS code?

If you are talking about a selection that is less than the entire line, there is no built-in way to duplicate selected text to the next line. It can be done with a macro extension which enables you to run multiple commands at once.

Using the macro extension multi-command try this keybinding (in your keybindings.json):

{
  "key": "alt+i",                        // whatever keybinding you want
  "command": "extension.multiCommand.execute",
  "args": {
    "sequence": [
      "editor.action.clipboardCopyAction",
      "editor.action.insertLineAfter",
      "editor.action.clipboardPasteAction",

      {                        // to add text after the selection
        "command": "type",     // you could also put this before the paste command
        "args": { "text": " myText here after paste " }
      }

    ]
  }
}

That will copy the selected text, insert a blank line after it and paste that text there. Demo:

Demo with adding static text to the duplicated text:

Click File > Preferences > Keyboard Shortcuts: Look for the Copy Line Down keyboard shortcut.