Can I use a header tag in footer inside section?
It's not valid HTML5 to have a &lt;header&gt; element inside a &lt;footer&gt; element. You could have checked this yourself on this site:

http://validator.w3.org/check

For using &lt;header&gt;, &lt;footer&gt; and &lt;section&gt; correctly, see this page:

https://developer.mozilla.org/en-US/docs/Web/HTML/Element/footer

For even more technical details, see the HTML5 specification:

http://www.w3.org/TR/html5/sections.html#the-footer-element

Special thanks to @amon for the links!