Welcome to the forum.
The REST API is capable of everything you’re asking about, but it’s quite low level, so you would need to in some cases make multiple calls to get all the information you want.
For a long running process you get the status by getting the process instance via rest If the it doesn’t return the details of the process then it either doesn’t exist or has finished.
You can get even more details on a process instance by asking for the history of the processes instance via rest. This will return information if the process is currently running OR if it has finished.