The government releases 500 spots for each day to enter to the Inca trail , these permits are for tourist, guides and porters, ever operator purchases as they have clients so basically is first come and firsts service, once the 500 permits is sold out there is nothing to do , even if there was any cancellation no one can take the spots.
The best months are from May to October, know as the dry season where you have amazing sunny days and blue sky at night
You can book huaynapicchu with Alpaca Expeditions at when you are booking your Inca trail.
Yes, do it on the four day after your inca trail, you will do circuit 3 which will take about 3 hours in total.
We suggest to be in Cusco 2 days at least before starting the hike.