I have the text styles shortcuts menu open. Is it possible to create/edit/delete these styles globally instead of just for individual documents?

(Mac Word 2011)

You can tell Word to apply the Style changes to the document's template as well but, whether that updates other documents based on the same template depends on whether those documents have the 'automatically update document styles' option checked. Either way, the update can only be 'automated' for documents based on the template of the document you're working with.

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Cheers,
Paul Edstein
[Fmr MS MVP - Word]

Some built-in styles cannot be deleted.
It is generally a bad idea to leave the option to update styles from the template turned on. See What happens when I send my document to someone else? Will Word mess up my formatting?