Apple Remote Desktop isn’t specifically needed here, and either it or the in-built screen sharing will work for what is a relatively minor part of what’s involved here. The bulk of this question involves getting dynamic DNS and security and the rest of the network connections working and secured. Or hosting the data on a shared resource; on iCloud or otherwise.
Probably the approach that’s easiest here is Back to My Mac, or one of the commercial remote-management offerings. If you’re inclined to set up your own remote access infrastructure, you’ll be learning about and implementing and troubleshooting firewalls and related topics. Probably also VPNs, as exposing file shares isn’t probably the best idea, and as any open network ports will be found and probed within minutes. And attacked.
Here are some recent discussions on the general remote-access topic, and will hopefully serve as some background. Have a look at these, and then swing back here with any remaining questions or confusions or concerns, and I’ll try to help further.
Re: apple remote desktop for parent's computer on another subnet?
Re: Can I Access and Manage my Mac mini at Home from my MacBook Pro While I'm at Work?
Re: Remote Desktop - computers in another state
Re: accessing another computer