How to verify if nginx is running or not?

This is probably system-dependent, but this is the simplest way I've found.

if [ -e /var/run/nginx.pid ]; then echo "nginx is running"; fi

That's the best solution for scripting.

You could use lsof to see what application is listening on port 80:

sudo lsof -i TCP:80

Looking at the requirement you have, the below command shall help:

service nginx status