You can access your website using only the IP address/port number like this: http://<IP address>:<port number>. I would suggest that you configure the web server to run on a port other than port 80, say 15231 or some other number of your choosing. As ISP assigned addresses are usually dynamic, you can register with a service that provides a domain name that works even when your IP changes. One such service is DynDNS.
As far as the search engines go, you can create a .htaccess file in your web root directory with the line “Header set X-Robots-Tag “noindex, nofollow” and this line will prevent your pages from showing up in the search engines.
If you set up your DMZ properly, other devices on your network should be safe. Your router/firewall documentation should tell you how to do that for your particular device.
You can set up authentication on your site and that will prevent anyone who doesn’t have the password from viewing it. You’ll use the .htpasswd file to do this. Here is an article
that describes the process in Apache web server.
You should be able to configure your router or firewall to block access from all addresses except the ones you will be using on the outside. Again, your router/firewall documentation will tell you how to do this.