you can see the winter solstice at Machu Picchu from Jun 21st to Jun 24th.
the llama sacrifice at Sacsayhuaman during Inti Raymi is definitely staged.
in the post-conquest history of Cusco, Corpus Christi and Inti Raymi are related but they are separate festivals. the first is a Catholic feast with moveable dates according to the liturgical calendar. the second is a "resurrection" of the Inca's winter harvest festival and is always celebrated on Jun 24th. the history of these festivals is quite fascinating - "Inka Bodies" by Carolyn Dean is a very good, tho very academic, book on the subject.
there is a special mass celebrated in Quechua in the Cusco cathedral on the morning of Inti Raymi.