I'll contribute what I can
Cost of renting: depends on how much space you need. I suggest a look at the classifieds in http://www.elcomercio.com.pe/clasificados/html/Cl-Home.html
to get an idea.
Cost of living: again, how long is a piece of string. Considerably lower than the U.S., except for utilities.
Cost of internet: various packages are available. Check http://www.telefonica.com.pe
, they're the biggest fone/internet provider with the greatest coverage.
What part of Barranco: As near the sea as possible. Barranco is very small, but the farther from the "downtown" you go, the less attractive. Between the Malecon and Av. Grau are the nice, colonial homes.
What to do in 3 days: wow, that's not near enough to do much more than get from the airport and back! Pick up a Sunday copy of El Comercio and start calling for viewings. The early bird catches the worm.