There are quite a few issues that I've found within the template files for the Network theme. I've gone through and fixed what I could while making notes about what I did so that I could share them here.
This will allow all pages I've tested at W3C Validator to validate. The only exception is default template page(s). I'm not sure can be validated without breaking things anyway .
It also fixes several layout issues with a default installation on the full width template. There are still some layout issues remaining on the full width and blog-news templates, but those are CSS related and I don't address those here.
Note: These fixes can all be accomplished using a child theme.
Code is at: http://pastebin.com/JNfxzHhj
Maybe those who know more than I can have a look and see what you think. I'm no WordPress guru so there may be better solutions for what I've done. (?)
Hopefully this will help others out =)