Are Tumblr themes updatable by author once used on a live site

From what I see you have "code" there; You can add a Syntax Highlighter script or something.

See here for a possible solution (it's in french, but you can use Google Translate)

Or, another thing I found: http://drnicwilliams.com/2007/03/08/syntax-highlighting-in-tumblr/

It may be tedious, but it looks like you'll have to include the HasTags block in each of the section types.

Starting with the first you'll end up with this:

    {block:Text}
       <li class="post text">
           <!-- HTML layout for Text posts -->

        {block:HasTags}
            {block:Tags}
                <a href="{TagURL}">{Tag}</a>
            {/block:Tags}
        {/block:HasTags}

       </li>
    {/block:Text}

Where you currently have it appear just before {/block:Posts}, change it to be just before the closing </li> which appears to be what your post container styles are dressed in.

Don't forget to remove the block that appears just before the closing {/block:Posts} or you'll get the tags showing up twice.