You could try using one of the existing Google Drive plugins to see if any are suitable for your needs. If you cannot find anything suitable in the WP repository or on the Internet at large, then you’re looking at a custom developed solution.
Google Drive does have an API that allows remote sites to interact with your Drive’s contents. I’ve not used it myself, but I’m fairly confident it would help you do what you want. The hard part is properly communicating with the API.
I would advise you not try to maintain a copy of Drive contents in your WP site. Always avoid redundant information, especially if keeping it in sync is important. It’s better to get any data you need dynamically, on the fly when needed, than to maintain redundant data. The API lets you do this.
You should keep all files that should be accessible in their own Drive folder. You could likely organize accessible and private data through permissions alone, but organizing by folder is much simpler and straight forward.
Thank you. I found this plugin worked very well.