BTW: UP/DOWN arrows and Backspace is redundant. Just use PAGE UP/DOWN or CMD LEFT (chrome, mac osx) for back.
At the very least the search input should be focusable by clicking tab ONCE... not 15 times.
EDIT: The whole Tab Order just makes no sense at all:
1. Try Prime 2. Sign In 3. Try Prime 4. Wish List 5. Shopping Cart 6. Departments 7. Fire & Kindle 8-14. No visible focus 15. Search input
I bet this has NEVER been A/B tested.
Click through the categories. You get much better results. (Not on mobile).
Autofocus is great on search engine homepages or search-oriented sites. Amazon isn't search-oriented when on the site. They assume their recommendations are usually spot on enough that you'll click through a percentage of the time, or that you googled straight into the product page (as most people do).
My favorite sites are the ones that don't autofocus, but have a keyboard shortcut (usually '/') that focuses on search. But then I love anything that behaves like vim/less.