Wrong document if script dynamically inserted into iframe head

Question 1

This behaviour is because, your jquery code is passing parent document object as context to Your google analytics snippet, so when Your google analytics snippet code is invoked function retrieve as parameter parent document object, you can check this behaviour using this example, You can see in dev tools in chrome on which document is pointing Your google script.

no_analytics.html

<!DOCTYPE html>
<html>
  <head>
    <style>body{color: #AAA;}</style>
    <script>
        console.log("document => ", document);
    </script>
  </head>
  <body>no analytics</body>
</html>

and index.html

<html>
<head>
    <script src="http://code.jquery.com/jquery-1.11.0.min.js"></script>
</head>
<body>
    <iframe id="insert" src="no_analytics.html"></iframe>
    <script>
        document.getElementById('insert').onload= function() {
            var c =
                "(function(i,s,o,g,r,a,m){

                  console.log('document from google script => ', s);

                 i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){"+
                "(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),"+
                "m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)"+
                "})(window,document,'script','//www.google-analytics.com/analytics.js','ga');";

            $('#insert').contents().find('head').append($('<script>').html(c));
        };
    </script>
</body>

To make Your example work I use javascript code without jQuery.

<!DOCTYPE html>
<html>
<head>
    <script src="http://code.jquery.com/jquery-1.11.0.min.js"></script>
</head>
<body>
    <iframe id="insert" src="no_analytics.html"></iframe>
    <script>
        document.getElementById('insert').onload = function () {
            var c =
                "(function(i,s,o,g,r,a,m){ console.log('document from google script => ', s);i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){" +
                "(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o)," +
                "m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)" +
                "})(window,document,'script','//www.google-analytics.com/analytics.js','ga');",
            documentIframe = document.getElementById('insert').contentDocument;

            var script = documentIframe.createElement('script');
            script.innerHTML = c;

            documentIframe.head.appendChild(script);
        };
    </script>
</body>

And here is the result of the above script

enter image description here

Question 2

I suggest you to try this:

$(window).load(function () {
    var c =
        "(function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){" +
        "(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o)," +
        "m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)" +
        "})(window,document,'script','//www.google-analytics.com/analytics.js','ga');";
    $('#insert').contents().find('head').append($('<script>').html(c));
});