No, it does start as root, but spawns child processes as a non root user (the one you have configured a user in the nginx.conf, generally it's www-data).
You can test this by starting nginx and then ps aux | grep nginx. You will see something like:
root 29757 0.0 0.1 140628 1504 ? Ss 11:21 0:00 nginx: master process /usr/sbin/nginx -g daemon on; master_process on;
www-data 29759 0.0 0.6 143300 6188 ? S 11:21 0:00 nginx: worker process
By default in /var/log/nginx. This location can be changed via config.
tail -f var/log/nginx/error.log
# or
tail -f var/log/nginx/access.log