1. I've found that when you enable the "Connect via on-premises data gateway" option, your security options are limited to "No authentication", "Basic authentication", and "Windows authentication", so the "API Key" option prevents me from entering the "Authorization" in the header. If you try to, you get an error indicating "Authorization" is not permitted in the header and to use the "API Key".
2. No gateways. I have several gateways for SQL, SharePoint, File System and FTP, but when I attempt to create a connection for the Custom Connector, there are no gateways. We have gone over the on premise gateways several times and cannot identify an issue.