Integrations should be fully built and tested before being connected to a live organisation. Once the integration is complete, you can hand it over to a Standard or Adviser level user to connect. Please see our Development Account page for ways to test your integration without cost.
The API essentially works on behalf of the user that authorised it to connect. Since the API acts with Standard user permissions, the user that connects the integration has to have at least Standard user permissions.
Generally apps using the API have the permissions of a Standard level user. To access reporting APIs the authorising user must have Reports access and for Payroll APIs the authorising user must be a payroll admin.
If your organisation isn't showing in the organisation dropdown, this means either that you don't have Standard or Adviser level permissions in that organisation, or you already connected that particular app to the organisation.