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.
There has been speculation that in a future Android version, Google will extend features of its Advanced Protection program to all devices. One of those is that sideloading of APKs will no longer be possible unless you do it over the command line via ADB. Only a tiny percentage of the population, techies like us, would be comfortable doing that. So, "you can still sideload" might not be a thing for long.
It definitely becomes tiresome than downloading an alternative chat app from Playstore. Signal has been gaining ground in India since Whatsapp came up with its new terms. A ban on Playstore will hinder that progress.
