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Something like: "With categories you can transpose problems across domains. Suppose you have a topology problem that's easier to solve in algebra. Well, you use categories to map the problem to algebra, solve it, and then map the solution back to topology." 
That's what algebraic topology does. But plenty of books on algebraic topology either don't mention category theory at all, or mention it only in passing. You can define homotopy, homology etc without categories, no problem.
yes and no. the illustration is about functors. and that is not domain specific. my sense is that the power of category theory is in extreme generality, but that is also why concrete, lay-friendly descriptions are hard. oh wait, i have an idea
sadly, i am some 10 years away from my once-decent knowledge of this subject, but there is a *very* simple demonstration of fibrations as a generalization of factor groups which i will now never be able to find or describe correctly. colleague showed it to me on acetate in berlin
The same way that black holes do not general relativity to exist 
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