User access and bank details

Can I restrict admin or finances users permissions to their own organisation?

Yes. Go to Setup > People, edit the admin/finance user's record, and tick the Local permissions only box.

How do I identify users who are missing bank details?

Set your default banking export format to Bacs Standard 18 (Setup > Settings). Once set, users missing required bank details are flagged with a warning icon (!) in the user list, hover over the icon to see exactly what's missing.