Beneficial proofs of work I'm new to this community and a Bitcoin novice. 

I believe to have understood the concept and purpose of proof of work but also, that the high energy consumption of proofs of work is a major problem of the Bitcoin (and other blockchain based) systems.

The puzzle that need to be solved for the proof of work has to be complex:


computing the cryptographic nonce (as used in the Bitcoin system)
solving complex differential equations
operations on dense matrices


I wonder if my impression is correct that these puzzles' only purpose is to provide authentication, esp. for the right to mine new bitcoins -  and that they are otherwise "useless".

More than that I wonder if there are attempts to define complex proof of work puzzles that fulfill other purposes that are more beneficial/useful/valuable to - let's say society - which would justify the high energy consumption.

You may compare this with Google's ReCAPTCHAs that next to authentication "assist in the digitization of books or improve machine learning".
Your first point is indeed right: The BTC puzzle is only useful for ensuring the security of the blockchain.

Onto your next question: One of the first attempts to define a "useful" PoW was Primecoin, created back in 2013. Additional research in this area is being done.

In defense of Bitcoin's PoW, some people have said that Bitcoin mining tends to use up energy that would otherwise go to waste. For instance, if a hydroelectric plant is no longer being used for its original industrial purposes, then it can instead be harnessed for BTC mining. Mining on fossil fuels is currently unprofitable.

Of course, a more "useful" PoW could harness this energy for something even better. But even then you have to weigh up the pros and cons of switching BTC/ETH to such a PoW function. And you'll probably come up short.