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Negative state prices in equilibrium can only occur if the utility of wealth is not always strictly positive. A natural assumption on utility is that it is strictly increasing in wealth in all states of nature (more money is always preferred to less money). Thus, unless you can come up with a situation in which an agent would prefer to be less wealthy and ...

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If you want to address interesting problems that are interesting for financial mathematics, I do not believe you have the good list. Pricing. For instance, most of explicit formulas for pricing that are not available yet will never be. In this direction, you should have a look at simulation techniques. See for instance Nonlinear Option Pricing. Interesting ...

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The whole machinery is needed to interchange the integral sign and the derivative.

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