It depends on your type of travel but if you are a backpacker so you can easily complete the trip under 8- 9k IND.
this is excluding flights. You can use a shared taxi from Jai Gaon to Paro. it will around 250/per person.
You just need to buy the form(will be available easily in the market) and go to the immigration office, your GOVT. The ID is required. it can be voter ID, Passport etc. Additionally, you will have to pre-book a hotel because you will have to submit your itinerary along with the prebook hotel booking.
I heard about Bhutan VISA is necessary for Indians as well, currently not required but not exactly don't know from when it is applicable. please check Bhutan official site before visiting.
Also, I get to know about this information from this Vlog below. you can watch it for more details and places to visit. as you said tiger nest, this guy has uploaded the trek video and his vlogs are informative. https://www.youtube.com/playlist?list=PLvFdD4FeLaivKj4fn6sKw7APtwHVFs6Ao