We use ballistocardiography (aka a big less sensitive microphone) to pick up movements, breathing and heart rate through the bed sheets.

Interesting. Has this been compared to EEG for accuracy? I can't imagine it to be very accurate. I'm really curious because as far as I understand anything that isn't EEG is pretty much useless.

Are the sound waves sent to the cloud for processing?

Nope, it's all filtered and processed locally.