At the moment Resthopper doesn’t really have a client-side API. You would have to serialize and pass along C# objects with RhinoCommon geometry inside. But this is a very interesting thought - could we potentially accept a JavaScript version of the DataTree with 3dm.io objects inside… Probably not in the nearest future, but that is definitely something for us to keep on the radar!
With regards to the third-party plugins, that depends on the server-side setup. In order to run, Compute needs an installed version of Rhino 7 (WIP), so basically, whatever GH plugins are installed on that machine are the plugins that you can call from Resthopper. The GHX file doesn’t really contain any executable code - it is just an XML markup that tells Grasshopper which components are located where, what connects to what and which assemblies each component corresponds to. The assemblies themselves have to be present on the server. @will will have a better idea of what additional plugins (if any) are currently available on compute.rhino3d.com.
If you do want to use plugins that are not available on the public Compute server, you can always install Rhino WIP, then clone the repo and spin up your own server. Rhino 7 can be downloaded from https://www.rhino3d.com/download/rhino-for-windows/wip and will share a license with your Rhino 6 installation. If you have any questions about the setup process, I’d be happy to walk you through.