Earlier quoted context omitted.
That's incorrect. All that matters is you can't game the system by taking shortcuts or deciding what the problem will be beforehand; the work cannot be even partially computable without knowing the previous block.
No, you are incorrect. The important aspect of proof of work is that it is very expensive to create an alternate history (maybe one where you didn't send $15,000,000 to the exchange, for example). If you are doing useful work, someone can pay for it, and you can use that payment to make an alternate history for free, defeating the security assumption of blockchains. Blockchains only work by probably wasting in order…
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.