Afaik, your questions are immediately answered by taking a quick look at the Revit Server end-user documentation, e.g., How Revit Server Works:
https://knowledge.autodesk.com/support/revit/learn-explore/caas/CloudHelp/cloudhelp/2022/ENU/Revit-C...
The first sentence answers your question 1:
> The Revit Server network architecture is designed to support worksharing of multiple central models across a wide area network (WAN).
So, the answer to question 1 is no, not exactly. You set up your own WLAN network to work with Revit Server. I guess you could call that a cloud, if you want, but I do not think that matches common usage.
Afaik, the answer to question 2 is yes: it is a REST API, so you can work with it using any programming language you like, cf:
https://thebuildingcoder.typepad.com/blog/2013/08/the-revit-server-rest-api.html
To answer your second question number 1:
> What are the options for running Revit in the cloud independently or via forge?
None, afaik.
Finally, according to my meagre knowledge, the answers to your last two questions are yes, in both cases.
Please also note that this discussion forum is dedicated to programming desktop Revit using the Windows desktop ,NET Revit API. For all Forge-related questions, you should ask through the dedicated Forge help channels:
https://forge.autodesk.com/en/support/get-help
I hope this clarifies.