Can I copy text containing Zotero citations into another Word document?

Yes, but in order to change styles and ensure citations are added to the reference list in the new document, you need to ensure the citations in the original document are still linked.

How to check if your citations are still linked:

Place your cursor on a citation (either in-text or in a footnote—the example below shows a footnote in Chicago style).
If the citation highlights with a grey background, the Zotero field codes are still present, and you can safely copy and paste it into a new document.

screenshot showing a footnote in Zotero which is greyed when the cursor is on it. This demontrates that there is Zotero code behind the text.

If the citation does not highlight with a grey background, it has been unlinked, and the Zotero code is gone. In that case, the pasted text will just be plain text, and you will need to re-enter the citations using Zotero. This is the only way to ensure Zotero can still update your citations and automatically generate a bibliography at the end of your work.

How citations become unlinked:

Citations are only unlinked if you select the 'Unlink Citations' option in the Zotero tab (Word menu bar).
Once this option is used, the Zotero codes behind the text are permanently erased, and the pasted text will appear as plain text.