Thanks! I did as you said and it worked for me. Still had some gotchas to figure out, before I got any satisfying results
I created a new role, especially for "web" context, duplicating "Load Only" role, and adding a new permission "Resource Tree". That solved the problem with no resource tree showing up, when pressing default "New Document" buttons in the resource tree or in the top menu.
I was also able to fix the top menu actions, by adding additional parameter "&context_key=en" to parameters list of each menu item. Still I can set only single "en" as the default context. Ideally I would make a new system setting "default_context" that could be set for individual users and add [[++default_context]] in parameters list, but that didn’t work out. I could try using javascript in the "Handler" field, but I don’t know how to read system settings using javascript. Is there a javascript API documentation for modx?
Hope this helps other people.