The root is relatively arbitrary, so as long as that’s the right location and structure, that’s all good.
Vladimir: Can you post your fastcgi_params file?
Also, it’s unrelated, but I’d recommend changing /blog/index.php?$args to /blog/index.php$is_args$args
When you use the straight ?$args version PHP thinks you’ve always got a query string (which is technically true, it’s just empty), so you can end up getting our “We saved your position when your session timed out!” notice every time you hit the admin, even if you didn’t have anything to save. $is_args will evaluate to a ? if there is a query string, otherwise it’s an empty string, which solves that problem.