Can I have two windows on the same Org-mode file, with different outline visibilities?
You may use indirect buffers which do exactly this. See the documentation on the command make-indirect-buffer. Org-mode has built-in commands that allow you create an indirect buffer from a subtree of an outline. To open a subtree in new window, type C-c C-x b. Any changes you make in the new window will be saved to the original file, but the visibility of both buffers will remain independent of one another. For multiple indirect buffers from the same file, you must use the prefix C-u when creating the second (or third) buffer. Otherwise the new indirect buffer will replace the old. You can also create an independent view of an outline subtree by typing b on an item in the agenda.