I think there are several issues here:
1. We receive a lot of emails several times, even days later. PUSH messages are also sent multiple times.
If you are receiving both emails and Push messages several times it could because there are two simultaneous cron / scheduled tasks running processing the same messages queue. Could you check in your server cronjob configuration if you are executing the scheduled tasks several times? This usually happens also in cluster environments, when you have more than one server or when a system was migrated from one server to another but the cronjob was not deactivated.
2. Debugging: Unexpected response from the Airnotifier server:
In this case what I feel is that the request failed because so many Push notifications were being processed at the same time for your site, it could be related to point 1 (duplicates emails and Push)
Apart from that, I checked the Airnotifier server logs and I didn't see any recent critical error reported by the server. In any case, we are going to add additional debugging in the Airnotifier responses so we can improve debugging in general.