There is no "nothing-at-stake" problem. No entity can ever buy 51% of the coins in existance.
The attacker needs 51% of coins the existed
at some point in the past. He can attack the network with 0% of the coins that are currently considered as valid. That's why it's called "nothing at stake": he can attack the network using coins that were already spent!
CPoS certainly fixes that.
My hunch is no. I think it just obfuscates the problem. DeathAndTaxes (and others like Gmaxwell, DannyHamilton, Grau, etc) are better at dissecting the fine details.
I, on the other hand, look at the problem from a macro perspective. My theory is that it is impossible to create an objective decentralized time-stamp server without a tether to the physical world. You see, proof of work translates a physical fact (energy expenditure) to a mathematical fact (entropy removed from block header). This is PoW's tether to the physical world. With PoS, all the rewards and expenses come from
within the system itself. It is my belief that it is not possible that such a system can remain objective with respect to physical reality.
I can't prove my hypothesis, however.