Hi,
Both point 1 and point 2 are probably because you didn't do "You have to specify this payment notification URL on your configuration account".
After a payment, the payment gateway (Sumup in this case) usually sends a payment notification to the website. For that the payment plugin integrating with the gateway either sends automatically the URL where the gateway has to send the payment notification, or asks you to manually provide that URL to the gateway. When the plugin receives the payment notification, it then changes the status of the order to confirmed and the email notifications are sent.
So this basically means that in the Sumup merchant account, there should be a place to provide that payment notification URL, and once you do so, it should all work fine.
It's probably what they call "webhook" here:
developer.token.io/sumup_rest_api_doc/co...ashboard/webhook.htm
Sumup's support should be able to help you with this.