The first thing you will want to do to remove the Active Directory domain mycompany.com and reconfigure AD to use a different domain name. Your AD domain name should be different than your public domain name so that these kinds of DNS issues don’t come up. Using an Active Directory name like mycompany.local will work just fine.
Second you should configure your servers to use private IPs and have the firewall use NAT to expose the servers to the Internet. You don’t want your servers just sitting on the Internet. Having them on the Internet will guarantee that they will be broken into.
Once these things have been done setup the database server’s DNS server to point to the server 1’s private IP address. Then you can add server 2 to the AD domain. Unless you plan on hosting a web site on the database server you don’t need to setup NAT for it at all. All public communication should go to server1, with that machine being the only one which talks to server2.