Can I, simultaneously, run two instances of the Facebook app? Can I, simultaneously, run two instances of the same Facebook app 
(which is installed once in the system)?

I want to be active on two accounts (I also have iOS and Windows Phone devices if their support for multiple instances of an app is better than Android's. But my Android phone is my main phone, hence I prefer an Android solution.) 

Also, can I get parallel notifications from both accounts (I don't need any chat messages, just the notifications from a specific Facebook group on my second account).
You cannot run two instances of app with same package name because when you try to install it out of the box because the package installer checks if an instance of the package you are trying to install is already present and if present asks if you want to update the existing app.  

You can however change the source code of an existing app to fool the system that it is some other app with a different package name  by cloning the source code and have a different package name. Reference link here



Also there's another way by making use of the special Facebook lite app.
 Facebook has provided a Lite app which is supposed to work for low-data-speed devices consuming less memory. 
You can install one Facebook native app and another Facebook lite app and use multiple accounts. So you'll be able to get notifications from both the apps.  

NOTE: This is possible because Facebook-app and Facebook-lite are two different apps (with two different package names com.facebook.katana and com.facebook.lite respectively)