Documents saved in a Zotero group, which belongs to a member with an unlimited subscription, can be visible to all members in the desktop version and everyone can work on them, modify them, etc?.

Yes. Though depending on what you mean by "documents", note that we don't recommend storing editable documents like Word documents in Zotero. File storage is meant primarily for PDFs and other research materials, and it lets you collaboratively annotate stored PDFs. If you try to store, say, Word documents that people have to save new complete versions of, you're going to end up with file conflicts and lost work if people are working at all simultaneously.

Would the documents be automatically downloaded to the desktop version?

It depends on the file-sync settings in the Sync pane of the preferences. You can set it to only download files as needed.

(But if by "documents" you just mean Zotero items, none of this applies and everyone in the group can obviously work on those without problems.)