In the sony connect app (or whatever it's called) you can also turn off the controls (or turn off just one side if you want). But I've also lain down with the headphones on and never had a problem with the pillow activating anything.

No, that's not possible, at least I haven't read that anywhere. I also sleep with the headphones on sometimes, because the input works by touch, input can be made through a pillow, the mattress etc. I don't know.