To your specific question - what are the implications?
Domain Controllers in a default configuration are not hardened for a public network, for example, they permit cleartext LDAP binds by default which might expose your passwords to interception. This article describes the process to disable LDAP simple binds https://support.microsoft.com/en-us/kb/935834
Depending on what you hope to achieve from a machine/user management perspective you should investigate the following technologies
Microsoft Intune can provide management of non-domain joined machines including Mac/Linux using Configuration Manager
Windows Azure Active Directory allows you to centrally create and manage user accounts and provide an ADFS authentication interface for various applications including Office 365.
DirectAccess allows for a Domain Joined experience while connected directly to the Internet by building an VPN tunnel to your cloud hosted network before authentication.
Workplace Join is a capability of ADFS that will permit you to "join" a device to your domain via the ADFS service.
Windows Azure can provide SMB shares over the Internet. But file shares are a legacy technology - use Sharepoint Online/OneDrive if you can.
Policies can (sort of) be done using Windows Intune - you won't get traditional Group Policy configuration, but you generally don't need that unless you want to lock down your environment.
Internet printing can be setup in Windows 2012 https://technet.microsoft.com/en-us/library/jj134159.aspx - but you would need a server somewhere for it. A cloud service undoubtedly exists.