If you want to piggy back on Google's centralized notification system (which saves network and battery usage), you have to distribute via the play store. I have no idea whether signal does this, but it would make sense for them to.
Signal has a means of delivering push notifications using their own infrastructure via websockets. It has a higher battery drain than using Google's push infrastructure, but it works.
You don't. You do need to create an app in the Google API console to obtain the key to send GCM notifications, but that's completely independent from the Play Store.
