Sadly, 8th gen consoles proved to the platform holders that hardware backwards compatibility is not as big of a seller anymore. Maybe the XBTwo will have BC through digital means, but I doubt the PS5 will have any at all, outside of the streaming alternative. And 3rd parties would rather make remakes/remasters at full price rather than allow people to already own them to continue playing them.
So no, they won't stop producing PS4s for a long time, maybe they will even reach the 100$ mark. XBOnes are going to stop production earlier, though.
From Mark Cerny interview PS5 would be compatible with PS4, it isn't a rumor.
duduspace11 "Well, since we are estimating costs, Pokemon Red/Blue did cost Nintendo about $50m to make back in 1996"
Mr Puggsly: "Hehe, I said good profit. You said big profit. Frankly, not losing money is what I meant by good. Don't get hung up on semantics"
Azzanation: "PS5 wouldn't sold out at launch without scalpers."