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Yes, you can set a custom URL slug for your bio page from Settings.
Yes — you can choose between light, dark, or “match device” themes for the badge, on either plan. This applies whenever the badge is shown: always on Free, and on Premium if you choose to keep it visible.
Yes, on Premium. Free accounts show a small “Powered by biogr.id” badge beneath your links; upgrading lets you turn it off.