It is possible to serve the Bitcoin Core GUI remotely via a web page running on a server running in Digital Ocean (or any other provider)? If so, where can I find instructions on how to do this?
It's not possible to use the GUI. (If you really want to, you can use TeamViewer, which is about controlling your computer remotely and not not that efficient.) You can use the RPC API to use the most of Bitcoin Core's functionality. If you're using it as a blockchain data API, the REST API might be a better choice.