Need some help with Public IP's please.

Hey pat,

It is hard to say what exactly are you trying to achieve with this sort of infrastructure. If you're trying to build a DNS service, why not to use Azure DNS... no necessity to reinvent the wheel. For mail sending you can use SendGrid (in Azure as well) which is quite simple SMTP relay. 

At any rate it all starts with Virtul Networks (VNETs), you need to create one, then use Azure DNS for resolving names in a VNET without the need for you to create and manage custom DNS solution. No necessity to explain that you can manage your own subnet segments and create custom private IP's that would be mapped to public IP. All the VMs inside the VNET would have an internet access by default. You can manage all the access via NSGs. 

If Azure DNS service is not an option, you can use names resolution for VNETs (through your own DNS server). https://docs.microsoft.com/en-us/azure/virtual-network/virtual-networks-name-resolution-for-vms-and-role-instances

To understand external IPs assignment and outbound routing - the best https://docs.microsoft.com/en-us/azure/load-balancer/load-balancer-outbound-connections?toc=%2fazure%2fvirtual-network%2ftoc.json#standalone-vm-with-no-instance-level-public-ip-address.

Hope this helps.