What do AVAILABLE, LEAVING SOON and OUT OF OFFICE mean, and can I set one directly?
They are the three statuses the app shows, and you cannot set one directly, through the interface or through the API: status follows from your rules, so to change it you create or change a rule.
You show OUT OF OFFICE while a rule is running, LEAVING SOON when a rule starts within the next 4 days, and AVAILABLE the rest of the time.
Related pages
Out of Office User View - where this is covered in full.
Rule Configuration - more questions on this topic.