Is any of them usable without excel e.g. on Linux?

There is no need to have Excel installed to use the OpenXML libraries.

https://github.com/OfficeDev/Open-Xml-Sdk

They also support being used on .NET Core.

The kind of API I was looking for but couldn’t find (in 2014) was “when I change the value of cell A1 to 17, what’s the value of cell C5?” And get the answer within a couple of seconds. There was no supported way to programmatically change the value of a cell and get the calculated result of a different cell, at least not that I could find.

http://github.com/ClosedXML/ClosedXML is a wrapper around OpenXML and attempts to fill that gap. It has partial support of Excel formula evaluation.

Disclaimer: I'm a current ClosedXML contributor.

Its weird to ask for an Excel API that doesn't use Excel. I guess you meant library.