1) How many btc wallets have been activated and how many of them are actively used?
Like steven0021 says, there's no way to know if the private key for a certain address is forgotten, or if the address is just dormant.
There are some statistics about the number of addresses containing inputs. You can generate your own statistics by using tools like
https://github.com/znort987/blockparser or import the full blockchain into a mysql database using ABE (requires weeks of your time and a sufficiently sized server). If you import the full blockchain into a mysql database, you could easily query how many addresses generated outputs or got imputs over the last x weeks.
By the way: 1 wallet usually contains dozens of addresses. An address that hasn't been used in years might still be part of an active wallet.
2) What is the emission of btc?Here I want to know if there are parked btc, or forgotten wallets.
12.5
BTC approximately every 10 minutes untill the next halving. I'm pretty sure a lot of people are hodling and lots of private keys have been forgotten over the years... The ratio between the two is impossible to figure out
3) What would be the fair prize for the one who gives the answer?
Nothing... since there is no right answer here.