Most of this process is documented really well by AWS here – https://docs.aws.amazon.com/apigateway/latest/developerguide/api-gateway-create-api-as-simple-proxy-for-lambda.html
The most important parts that I believe are missed are when it comes to using the custom domain name. You have to set up a CNAME record to the API Gateway domain name shown in the Custom domain names section.
You do NOT want to point your CNAME to the test URL provided. Also…the stage is only used in the test URL, not the actual one used in your custom domain name API mapping.