How/Does it handle push notifications to iOS/Android/Web? Similar to Firebase?
Similarly, look to Amazon Cognito for user sign-in / sign-up / etc., AWS Lambda and API Gateway for cloud-based API, and other services for machine-learnign, AI, chatbots, database services, analytics, etc.
AWS AppSync is only for the data access / offline / real-time component.
Is this using AWS IoT?
I don’t know.
I don't know, but when using AWS, SNS is the push based solution.