Blockchains only work by probably wasting in order to build a history. If you aren't wasting, then it wasn't expensive to build the history (it was subsidised), and it wouldn't be expensive to build a different history (that one could also be subsidized)
It's possible for something to provide a social benefit but not make sense for anyone to pay for as a for-profit business because others would get the same benefit and free ride off them. (Hence the problem of public goods in economics.)
So if the proof of work is only solving some general research problem, then it might not be feasible to double dip like you've described. That's why primecoin was able to avoid the problem.
If you attack the network you can steal billions - so if you have that much computing power available for your useful work, you can temporarially use it against the block chain.
As long as that mining power is available anywhere the chain isn't secure.
