Although Zotero automates many processes for you, the output isn't always 100% correct. You will need to be familiar enough with your style so that you can verify and fix the formatting in Zotero before you use the citations in your document.
Use the Citation Guide along with Zotero
No, the Zotero desktop program and Connector work on your laptop or personally-owned computer, not on NPS-networked computers.
When on networked computer, you can add and edit items in your online zotero.org library, but you cannot use it to cite in Word.