Best season to visit Gokarna would definitely be from October to March. You will find each activity operational and could enjoy delicious food at beach shacks.
In my opinion .. its all the year round. I had visited in peak summer( you can check my posts) , it was still pleasant. The best would be winters, but thats the time when it would be the most crowded. If you want a budget friendly experience, go for slightly off beat times, but do avoid moonsoons, because it would mean you cant enjoy the beaches, well, then what would be the point of visiting Goa then!
EBC (Everest base camp) wil cost you around 1- 1.25 lacs.
You can plan your trip on your own ,where your budget will be reduced something around 20k
Bt it's advisable to organise through trip organizers..
Best month for EBC wil be May or June
There is no official website to visit dudhsagar waterfall the only way is to reach entrypoint as early as possible so that you can avoid crowd during visit. second is go through train and then trek but i am not sure now it is allowed or not.