Mac Mini 2013 remote setup with a macbook pro

If you only want to host VNC on your Lion server I'd suggest opening the specific port for that service (TCP 5900) on your modem/router and forward it to your Lion server, as opposed to putting it on a DMZ which fully exposes the machine to the internet.

To be able to get to your server using a human readable address like - server.mydomain.com, you need to own a domain (mydomain.com) and then link your actual internet IP address (that your internet provider gives you) to that domain (e.g. server.mydomain.com). It's pretty much as simple as that if your internet IP address doesn't change (is static). If your IP address regularly changes, this kind of approach isn't really possible...

The easiest way to get a DNS (human readable) address is to use a service like DynDNS or No-IP. These constantly check what your IP address is and keep internet DNS records up to date with your current IP address. It requires that your modem/router is compatible with such a service, or that you run a piece of software on a machine (your Lion server maybe) that's inside your network.

So;

1) Forward port 5900 TCP to your Lion server (Port forwarding settings in your modem/router)

2) Set up DNS so that you don't have to remember your IP and so that if your IP changes, you can still connect back to it.

3) Enable screen sharing on the Lion server. Tell it which users you want to be able to connect. Set a password.

I'd recommend looking into using your Lion box as a VPN or SSH server so that you can do all these kinds of things over a single encrypted link. VNC isn't the safest of protocols!

Boot the mini while holding down the Option (Alt) key. It should give you a menu where you can select the Lion Recovery partition. From there, you can reset the passwords or (if needed) reinstall the OS.