Is it possible to recover apps removed from Google Play? I recently discovered that on my new phone (the old one died suddenly because of eMMC failure) some apps are no longer on Google Play. They are not incompatible with device; they are not listed at all.
I have some old backups by Helium Backup, but I am 100% sure that newer versions were released. 

Is it somehow possible to download these apps? Most of them I purchased, but I can't find them even in "Library". I contacted their respective developers, but got no answer at all...
You can try to find the app on a site such as apkmirror.com or apkpure.com. Download the .apk file for the app, and install it using adb. This is called "sideloading," and you can find instructions for it here or here.