What happens in case of a hard fork? How will they ensure that no one will validate blocks in the wrong fork? With PoW the miners much choose one fork or at lest split the hash power. With PoS the miners can validate blocks in both hard forks without too much effort.
Moreover, if the 51% dislike the one fork, or the version before the fork, it's too easy to validate correctly in the part of the fork they like and launch a 51% attack to the other fork. This would have killed ETC and BCC instantly. (I'm not sure if this is a bug or a feature.)
That is only true for coins that are recognized on both chains. In the event of a hard fork, all you need to do is send your hardforked coins to a new address, and then start staking from that address. You can then vote on both chains with your coins, and no penalty algorithms will punish you.
If there is a hardfork and you're staking then I'd imagine you'll also be automatically staking on both chains, nothing to do. Hardfork only matter if you're transacting at the time of the fork, your transaction could be replayed on the other chain or if a reorganisation happens, in that case you may want to wait until the fork gets resolved.
Not quite. Mining on the wrong chain doesn't penalize you, it's mining on multiple chains that does. Only a bad actor would mine on two chains simultaneously.
Moreover, if the 51% dislike the one fork, or the version before the fork, it's too easy to validate correctly in the part of the fork they like and launch a 51% attack to the other fork. This would have killed ETC and BCC instantly. (I'm not sure if this is a bug or a feature.)