You call it just like you call any other function. Your handler takes in either a JSON payload and a context object or a deserialized object depending on the language. You call it from a separate console app, your test harness etc.

Yup. You can log into the AWS console and generate an example test message. Copy that into your unit/integration test and bob's your uncle.

We deploy to a test stack and run a full integration test of the code running in AWS. I believe it's also possible to self-host locally, but we never really looked into it.