I'll take a crack at this, since I applied for MPP programs. (For those curious, I got into Chicago with $, turned down at Kennedy School).
Public policy and political economy are very, very different! With either you could teach or do research, but policy programs are more practical-oriented, while political economy would be more "academic," with more theory and research.
What are the programs like? It really depends on the program. Some, like George Washington University for example, are part-time and very practical. Many of their students enroll part-time while working in DC. Others, such as the University of Chicago's Harris School, place an emphasis on quantitative skills and research.
Could do anything. People I know with MPA/MPP/policy PhD degrees work for non-profits, in business, in the government, and in journalism. PhD opens the door to more serious research and teaching, but a master's is also valuable.
Please think about taking some time off, and not going straight into the program. Make sure this is what you want to do. Most people in policy programs will have practical experience, and adcoms also look for people with real-world experience. I think the program will also be more valuable to you if you can see how what is learned in the classroom applies to the real world.
Everyone has their own path, and schools will respect that not everyone can afford to work for free every summer in Congress. However, of course something that is related to politics or policy/research would certainly help, or anything that demonstrates your commitment to public service.
Everything.... Though essays and LORs probably matter much more than in, say, hard sciences or economics.
Take time off! Get some experience outside the Ivory Tower. Talk to people with those degrees. And also talk to people whose jobs you might someday want to have, and ask them how they got where they are.