Is it correct that Firebase in-app messages will work even when push notifications are switched off or not permitted?

Yes.

I know how to send Firebase in-app messages using Firebase Console. Can I do it from my back-end?

No.

Using API or AWS SNS?

No, there is no documented API for this. If this is something you want, you should contact Firebase support directly to make a request. Stack Overflow can't help with this.