Your temp estimates for Lima are pretty accurate although the Peruvians, like the rest of the world, talk Celsius, so expect around 15C at night and highs of around 20C. There will probably be a lot of low marine cloud which may or may not burn off in the afternoon, familiar conditions if you know coastal southern California in June. Humidity levels are high day and night.

For the other sites you're pretty close as well. You should definitely have a pullover or fleece for Cuzco and the early morning train ride to Machu Picchu. You should also be prepared for some rain. September is a "shoulder" month in the Andean cordillera, during the transition from the dry season to the rainy season. If my past experience is any guide conditions will be more dry than wet but the rains are variable from year to year, so it doesn't hurt to take goretex with you.

As far as shorts are concerned, even in the jungle many Peruvians are more formal than North Americans, and favor long pants. In Lima I can practically guarantee that you won't see anyone wearing shorts, even if you go to the beach (because there will be nobody there).