Ubiquitous has provided a very good explanation for what constitutes understanding an economic problem. I'd like to address the second part of your question about what sort of "key questions" are solved in economics (if any).
First, the obvious. We have to talk about what meaningful economic problems are, that economists are best suited to address. This is my attempt at structuring the major issues (and there will probably be disagreements with the importance I place on some aspects).
Fundamental Problem: Scarce resources and unlimited human desire requires us to determine:
- What to produce and to do it efficiently, that producing more of a good does not reduce the production of another.
- How to allocate the goods efficiently, so that no one can be made better off without making someone else worse off.
Public Problem: Externalities absolutely reduce welfare. Public goods and services will not be allocated efficiently under normal market conditions, which absolutely reduces welfare, and require us to determine:
- How to find the least costly/burdensome way of internalizing externalities (including deadweight loss).
- How to end up at efficient allocations after subsidies and taxes, so that no one can be made better off without making someone else worse off.
- How to balance the social planner's public budget among infinite periods. (The state is not corrupt and increasing its welfare at the expense of others.)
Welfare Problem: Macroeconomic volatility absolutely reduces welfare. Poverty and persistent, large income inequality have negative externalities that reduce total social welfare. (You could also argue that poverty should be treated as a public good and so falls under the previous category.) These require us to determine:
How to set macroeconomic stability policy efficiently, where the tradeoff between inflation and growth cannot be improved upon. We may be willing to improve one aspect at the expense of the other though.
How to provide insurance against risk efficiently and broadly, where everyone is able to consumption smooth by accessing insurance that guarantees a basic level of welfare for themselves.
How to guarantee a basic level of welfare for every user through a balanced budget tax or subsidy with the lowest deadweight loss.
The last section is probably the hardest for me to be precise about. Is poverty about absolute standards, relative standards, or being most at risk to suffer greatly from unemployment? What is everyone's marginal cost for eliminating poverty (especially problematic since it is probably a function of actual income)? Nevertheless, I think it's some of the most compelling material we have to deal with in economics; the reason we care about all this is because of human interest, and this is where we'll find the most competing interests.
So what are some key questions in economics?
Here are some common, relevant ones, that are for the most part solved:
- Are price controls welfare reducing? (Yes)
- Are tariffs on trade welfare reducing? (Yes)
- Do large federal deficits hurt a nation's economy? (Yes)
- In an ideal economy, should we have floating or fixed exchange rates? (Floating)
- Are metal backed currency standards garbage? (Yeeeeessssss)
Those are a few that I first think of when writing this. But there are plenty more that are unsolved:
- How do we resolve the equity-premium puzzle?
- When should a government subsidize trade to resolve the Myerson-Satterthwaite bilateral trade problem?
- Is there really a feasible Walrasian auctioneer for [insert market here]?
Maybe these questions seem very small scale compared to other questions you posed above, but they serve as important foundations for understanding further ideas, or for the unsolved ones, they serve as long standing problems that probably require a whole new innovation to grapple with.