Question

Data sync is NOT enabled

Can I add hidden record fields if data sync is NOT enabled on a record type?

If by hidden, you mean unmapped fields, the answer is no.

By hidden I mean add fields to a record type that do not appear in the source database. And I could add custom record fields?

Fields that are in the Record but no in the database are Custom Record Fields. And those require data sync to be enabled.

Here's the full list of features you get with Data Sync:

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